It would be too awkward for him
Harry does not seem to know how to deal with the fact that his friends are poor. Throughout the series they almost never discuss money, even though Harry spends literally every waking moment of his life with Ron. The few times that money does come up, we find Harry feeling awkward and not knowing what to say or do. For example, in Chamber of Secrets when George expresses concern as to how the Weasleys will afford everything for the new year, we find Harry's response (or, rather, non-response):
Harry said nothing. He felt a bit awkward. Stored in an underground
vault at Gringotts in London was a small fortune that his parents had
left him.
Later when he goes to Gringotts with the Weasleys we see his embarrassment at the fact that he has money and they don't:
Harry enjoyed the breakneck journey down to the Weasleys’
vault, but felt dreadful, far worse than he had in Knockturn Alley,
when it was opened. There was a very small pile of silver Sickles
inside, and just one gold Galleon. Mrs. Weasley felt right into the
corners before sweeping the whole lot into her bag. Harry felt even
worse when they reached his vault. He tried to block the contents
from view as he hastily shoved handfuls of coins into a leather bag.
All the times Malfoy makes fun of the Weaselys' poverty, Harry doesn't seem to know how to respond. While he might help fight Malfoy he doesn't actually deal with the underlying tension. In fact, in Goblet of Fire we are expressly told that Harry didn't know how to deal with it (my emphasis):
“Must be nice,” Ron said abruptly, when they had sat down and started
serving themselves roast beef and Yorkshire puddings. “To have so
much money you don’t notice if a pocketful of Galleons goes missing.”
“Listen, I had other stuff on my mind that night!” said Harry
impatiently. “We all did, remember?”
“I didn’t know leprechaun gold vanishes,” Ron muttered. “I thought I
was paying you back. You shouldn’t ’ve given me that Chudley Cannon
hat for Christmas.”
“Forget it, all right?” said Harry.
Ron speared a roast potato on the end of his fork, glaring at it.
Then he said, “I hate being poor.”
Harry and Hermione looked at each other. Neither of them really knew
what to say.
In Order of the Phoenix when explaining Percy's defection, it seems like Ron can't even bring himself to explicitly say that they are poor (my emphasis):
“He went completely berserk. He said — well, he said loads of
terrible stuff. He said he’s been having to struggle against Dad’s
lousy reputation ever since he joined the Ministry and that Dad’s got
no ambition and that’s why we’ve always been — you know — not had
a lot of money, I mean —”
On the rare occasions when Harry does buy things for the Weasleys he does it awkwardly, as in Chamber of Secrets when he gives Ginny a set of books (my emphasis):
“You have these,” Harry mumbled to her, tipping the books into the
cauldron. “I’ll buy my own — ”
Or he jokes that that it is instead of many Christmas presents, as in Goblet of Fire when he buys omnioculars for Ron and Hermione:
“No — don’t bother,” said Ron, going red. He was always touchy about
the fact that Harry, who had inherited a small fortune from his
parents, had much more money than he did.
“You won’t be getting anything for Christmas,” Harry told him,
thrusting Omnioculars into his and Hermione’s hands. “For about ten
years, mind.”
(This last quote also provides further evidence of "touchiness" with regard to the money situation.)
Finally, when Harry wants to get Ron new dress robes (in Goblet of Fire), he explicitly tells Fred and George to say that it's from them:
Buy Ron some different dress robes and say they’re from you.”
In other words, it is too awkward for Harry to simply buy Ron new robes himself.
We do in fact find at times that Harry is somewhat impetuously reckless with his money. For example, after getting acquitted in Order of the Phoenix:
With a grin at the thought of what Hermione would say if she could
see the statue of the elf, Harry turned his money bag upside down
and emptied not just ten Galleons, but the whole contents into the
pool at the statues’ feet.
Therefore, while it is perhaps believable Harry might be willing to waste a lot of money on brooms for the rest of the team, or even just for Fred and George, it seems likely that he would simply be too uncomfortable to do so.
While Harry does give Fred and George a thousand galleons at the end of Goblet of Fire, that situation seemed to be less of a "charity case" and more of Harry wanting to get rid of the money from the Triwizard Tournament, and figuring that funding a joke shop would be a good use for it.
“Listen,” said Harry firmly. “If you don’t take it, I’m throwing it
down the drain. I don’t want it and I don’t need it. But I could do
with a few laughs. We could all do with a few laughs. I’ve got a
feeling we’re going to need them more than usual before long.”
Additionally, as we see in the beginning of Half-Blood Prince, the Weasley twins seem to have viewed this more as a loan than as charity:
“You gave us our start-up loan, we haven’t forgotten,” said George
sternly. “Take whatever you like, and just remember to tell people
where you got it, if they ask.”
The very fact that they insisted that it was a loan, and sort of tried to pay it back, might be further indicative of the awkwardness of receiving money from Harry.
