It's tempting to argue from a position of gentrification, but I actually have a better argument: the brooms don't seem to confer any great advantage.1
Let's look at the Gryffindor v. Slytherin Quidditch record, as far as we know it.
Philosopher's Stone2
Gryffindor wins 170-60:
"He didn't catch it, he nearly swallowed it," Flint was still howling twenty minutes later, but it made no difference - Harry hadn't broken any rules and Lee Jordan was still happily shouting the results - Gryffindor had won by one hundred and seventy points to sixty.
Philosopher's Stone Chapter 11: "Quidditch"
Since the Snitch is worth 150 points, this means that Slytherin scored 6 times, while Gryffindor only scored twice.
Chamber of Secrets
The game is interrupted by Dobby attacking Harry with bludgers, but the score before he goes down is 60-0 in favour of Slytherin:
It had started to rain; Harry felt heavy drops fall onto his face, splattering onto his glasses. He didn't have a clue what was going on in the rest of the game until he heard Lee Jordan, who was commentating, say, "Slytherin lead, sixty points to zero - "
Chamber of Secrets Chapter 10: "The Rogue Bludger"
We never hear any indication that either side scored again, and the time difference between when the score is announced and when Harry goes down is quite small, so I'm inclined to say that the final score was 150-60 in favour of Gryffindor; Slytherin scores 6 times, Gryffindor scores zero.
Prisoner of Azkaban
Finally, Gryffindor is able to out-score Slytherin; the final score is 230-20:
"[Angelina] SCORES! SHE SCORES! Gryffindor leads by eighty points to twenty!"
[Later, Harry catches the Snitch]
Prisoner of Azkaban Chapter 15: "The Quidditch Final"
So Gryffindor scores 8 times, Slytherin scores twice. Unusually poor showing from Slytherin this year.
Order of the Phoenix
Final score 160-40 in favour of Gryffindor:
' - and Katie Bell of Gryffindor dodges Pucey, ducks Montague, nice swerve, Katie, and she throws to Johnson, Angelina Johnson takes the Quaffle, she's past Warrington, she's heading for goal, come on now, Angelina - GRYFFINDOR SCORE! It's forty-ten, forty-ten to Slytherin and Pucey has the Quaffle... '
[Later, Harry catches the Snitch]
Order of the Phoenix Chapter 19: "The Lion and the Serpent"
Slytherin scores four times, Gryffindor scores once.
Half-Blood Prince
This one is a rout. The final score isn't clear, but it's at least 250-0 for Gryffindor:
With half an hour of the game gone, Gryffindor were leading sixty points to zero, Ron having made some truly spectacular saves
[...]
It seemed as though Gryffindor could do no wrong. Again and again they scored, and again and again, at the other end of the pitch, Ron saved goals with apparent ease.
[...]
Harry accelerated; the wind was whistling in his ears so that it drowned all sound of Smith's commentary or the crowd, but Harper was still ahead of him, and Gryffindor was only a hundred points up; if Harper got there first Gryffindor had lost
Half-Blood Prince Chapter 14: "Felix Felicis"
This game is easily Gryffindor's best overall showing (Harry notwithstanding), and it's largely thanks to Ron's preternaturally good Keeping.
What's my point?
Comparing the score count between Philosopher's Stone and Chamber of Secrets, it's clear that the Nimbus 2001's don't give the Slytherin team much of an advantage; they score just as many times as they did before they had them, and really all they can claim is that they prevented Gryffindor from scoring at all3.
The fact of the matter seems to be that, faster brooms or not, the Slytherin team is just generally better than the Gryffindor one, although there's obviously some variability as players cycle through; all Gryffindor really has is Harry, who routinely outclasses the Slytherin Seeker regardless of what brooms they're riding, and on one occasion they have a totally in-the-zone Ron as a Keeper. Skill counts for a lot more in this game than it initially appears.
But everyone says they do
This is a fair counter-argument; as the question points out, people always seem to react as though having the Latest Broom is a massive benefit to the other team. There are two examples in Chamber of Secrets, first when the Nimbus 2001's are first revealed:
All seven of [the Slytherin Quidditch team] held out their broomsticks. Seven highly polished, brand-new handles and seven sets of fine gold lettering spelling the words Nimbus Two Thousand and One gleamed under the Gryffindors' noses in the early morning sun.
"Very latest model. Only came out last month," said Flint carelessly, flicking a speck of dust from the end of his own. "I believe it outstrips the old Two Thousand series by a considerable amount. As for the old Cleansweeps" - he smiled nastily at Fred and George, who were both clutching Cleansweep Fives - "sweeps the board with them."
None of the Gryffindor team could think of anything to say for a moment. Malfoy was smirking so broadly his cold eyes were reduced to slits.
Chamber of Secrets Chapter 7: "Mudbloods and Murmurs"
And during the actual match, Harry observes that the Slytherin team seems to be gaining an advantage from their new brooms:
It had started to rain; Harry felt heavy drops fall onto his face, splattering onto his glasses. He didn't have a clue what was going on in the rest of the game until he heard Lee Jordan, who was commentating, say, "Slytherin lead, sixty points to zero - "
The Slytherins' superior brooms were clearly doing their jobs
Chamber of Secrets Chapter 10: "The Rogue Bludger"
I have three counter-counter-arguments:
All the characters we hear from are kids. This may seem like a facetious argument, but it's not; all the praise we hear about fancier brooms comes from twelve-year-olds, and thirteen-year-olds when Harry gets a Firebolt in Prisoner of Azkaban. It wasn't that long ago when I was that age, and I can say with some authority that children tend to conflate technical ability with skill.
None of the people we hear from are unbiased. The people who we tend to hear expounding the virtues of better brooms are (in roughly decreasing order of frequency) Oliver Wood, Ron, Malfoy, and Harry. Every single one of these characters has their blinkers on. None of Oliver, Harry, or Ron want to admit that Slytherin might be better than them at anything, so they're actively looking for alternative explanations. Malfoy, on the other hand, assumes that everything about him (and Slytherin) is better than Harry or Ron (and the Gryffindors), and he's actively trying to antagonize them
BlueRaja - Danny Pflughoeft comments:
Everyone says that about real-world bowling balls, golf clubs, professional gaming mice, etc. In reality any advantage is miniscule in comparison to the skills of the player.
Which is a very good point. It's not remotely uncommon among any kind of hobbyists to put a lot of stock in their tools, and argue over which one is "better." Whether one actually is objectively better than another is irrelevant4
Professor McGonagall is specifically mentioned in the question, so I want to address her opinion on the brooms for a moment. She's the first advocate of Harry's Quidditch career, but she doesn't recommend Harry's broom; Oliver Wood does (emphasis mine):
"He's just the build for a Seeker, too," said Wood, now walking around Harry and staring at him. "Light - speedy - we'll have to get him a decent broom, Professor - a Nimbus Two Thousand or a Cleansweep Seven, I'd say."
Philosopher's Stone Chapter 9: "The Midnight Duel"
The only thing McGonagall has to say about the broom is the note she attaches to the package, which contains very prudent advice but no discussion as to its merits:
DO NOT OPEN THE PARCEL AT THE TABLE.
It contains your new Nimbus Two Thousand, but I don't want everybody knowing you've got a broomstick or they'll all want one. Oliver Wood will meet you tonight on the Quidditch field at seven o'clock for your first training session.
Philosopher's Stone Chapter 10: "Hallowe'en"
1 Based on the extended discussion in comments, I want to clarify that I'm only talking about the amateur Hogwarts Quidditch league, not the professional league. That's really what the questioner is asking about, and we don't have enough information about professional Quidditch players to say how much of an impact equipment really plays. It could be a lot, it could be a little; we don't have any canon basis.
2 Sorcerer's Stone to you American heathens
3 In comments, Lilienthal questions whether I'm justified in making this comparison. There's certainly a point to be made that you need to be careful in these sorts of things, because the impermanence of the teams' rosters gives you another variable to consider: is this relationship due to equipment, or because one team got an excellent player last year, or some combination?
However, I feel like I'm justified in this case. The Quidditch teams (at least Gryffindor and Slytherin's) don't change very much:
Slytherin changes very infrequently, and in particular the only difference in the roster between Philosopher's Stone and Chamber of Secrets seems to be Malfoy
Gryffindor changes even more rarely; the roster stays exactly the same until Order of the Phoenix, when Oliver Wood (who graduated at the end of Prisoner of Azkaban, but there was no Quidditch in Goblet of Fire) is replaced with Ron. The roster does shake up dramatically in Half-Blood Prince (only three first-string players kept from the previous year, two of them dating back to Philosopher's Stone), but they get dramatically less experienced
4 It doesn't just apply to sports either. Go onto a PC forum and post about how you prefer Macs, or vice-versa, then duck. These are often examples of choice-supportive biases (especially post-purchase rationalization or, as I prefer to call it, "Buyer's Stockholm Syndrome")), where we basically decide that one thing is better than all the others because we bought it.