And now for something completely different...
I'm going to go against the accepted wisdom here and say that Professor Trelawney's possession-type prophecies are not the only valid divination techniques shown in Harry Potter.
Professor Trewlaney correctly predicts the events at the tower in Harry Potter and the Half-Blood Prince
Let's see what she says:
“The headmaster has intimated that he would prefer fewer visits from
me,” she said coldly. “I am not one to press my company upon those who
do not value it. If Dumbledore chooses to ignore the warnings the
cards show —” Her bony hand closed suddenly around Harry’s wrist.
“Again and again, no matter how I lay them out —” And she pulled a
card dramatically from underneath her shawls. “— the lightning-struck
tower,” she whispered. “Calamity. Disaster. Coming nearer all the time
. . .”
Certainly the Tower, as the Tarot card is more commonly known, is a generic indicator of catastrophe (among other things), and there is little that Professor Trelawney loves more than to predict catastrophe. But the choice of this method, of all things, and the Tower as referent (when indeed Dumbledore dies on the tower) seems to me to be more than coincidence. As stronger evidence, the very next chapter is called "The Lightning Struck Tower." No prizes for guessing what happens here. I suspect Rowling is giving us a not-so-subtle clue about the accuracy of Trelawney's prophecy.
Indeed, if we are to take Trewlaney at her word here, she continued getting the Tower no matter how many times she laid the cards out, an event which certainly defies probability.
She also detected Harry as the object of her predictions
“Two of spades: conflict,” she murmured, as she passed the place where
Harry crouched, hidden. “Seven of spades: an ill omen. Ten of spades:
violence. Knave of spades: a dark young man, possibly troubled, one
who dislikes the questioner —”
She stopped dead, right on the other
side of Harry’s statue.
“Well, that can’t be right,” she said,
annoyed, and Harry heard her reshuffling vigorously as she set off
again, leaving nothing but a whiff of cooking sherry behind her.
She detected a "dark young man" (meaning dark-haired), "troubled" (naturally), "one who dislikes the questioner" (which pretty much sums up Harry's attitude toward Trelawney.). That the situation is played for broadly comedic effect should not distract us from the fact that this whole scene is very obviously written to show Trelawney casting tarot for Harry, while being unaware that she is doing it.
This is the key, by the way.
When Trelawney falls into her trances, she is unaware that she is prophesizing. She is no more aware when she uses the actual tools of divination. Professor Trelawney is consciously a fraud, but unconsciously a seer, not only with her trances but with other divinatory techniques. Notice how in the first case, she admits to laying out the cards again, and in the second, she shuffles them when they fail to give the result that she desires. She tries her best at being a fraud, but her natural talents shine through.
Sybill Trelawney is clearly a genuine seer. Why should we be so surprised if there is more than one way to make a correct prediction?
Many other people have noted that a surprising number of nonspecific prophecies made by various characters came true
For example, Trelawney "predicted" Hermione leaving the class, or Lavender Brown's rabbit being eaten.
Harry "predicted" that Buckbeak would escape.
And how can we forget one of Harry and Ron's few mentioned divination homework sets:
On Monday, apparently, Harry will be in danger of burns.
On Tuesday, he will lose a treasured possession.
He will then be stabbed in the back by someone he thought was a friend.
He will lose a bet and come off worse in a fight.
The first two "predictions" correspond well to the first two tasks. Of course, he is later betrayed by Ron or, more plausibly Crouch-as-Moody. Later Bagman loses a bet he has on Harry to win the tournament.
Though each of these as individual evidence is highly suspect, together they might provide statistical indications that the average wizard can make predictions that are, at least, more likely to be correct.
We also have more direct evidence.
From the Famous Wizard cards, Calchas and Mopsus competed to see whose Divination was better. Since these cards were written by Rowling, they can be considered canon. I suspect that "competing" does not plausibly mean "both waited until they had visions and then discussed them." Rather, they used divinatory techniques. Of course, one could argue that these cards are not written from an omniscient authorial viewpoint.
The fact that Divination is taught as a legitimate subject is telling.
I find it unlikely that wizards would teach divination as a subject if it were completely ineffective. So many of their spells are flashy and immediate, that the suggestion that they would adopt as a subject something as subtle as divination with no payback seems a little implausible.
This probably has some relation to Rowling's statement that she could not see Wiccans as feeling comfortable at Hogwarts: with the more immediate and obvious effects of wizardly magic, paganism and such would seem less plausible. (I actually disagree; a modified form of Wicca centered around the magic that wizards normally do could be quite successful). If Rowling finds it implausible that Wicca would be plausible in a world of instant-gratification magic, I find it hard to believe that fortune-telling would be any different.
Far more likely that divination is a subject of some use at least to Seers (as evidenced by Trewlawney's accurate predictions), and perhaps of some use to wizards generally, but prone to error. Far more likely that Professor Trelawney is simply a particularly bad Seer, who cannot use her powers very well. Clearly Cassandra Trelawney made prophecies more frequently, at the very least.
According to Pottermore:
Sybill is the great-great granddaughter of a genuine Seer, Cassandra
Trelawney. Cassandra's gift has been much diluted over ensuing
generations, although Sybill has inherited more than she knows.
Nevertheless, Sybill does experience very rare flashes of genuine
clairvoyance, which she can never remember afterwards.
J.K. Rowling could not have too many accurate prophecies, as discussed in her interview. This could also be viewed as a plausible justification for the relative ineffectiveness of divination as taught at Hogwarts.
Keep in mind, though, that Professor Trelawney's gift is, short of large-scale centaur astrology, the only direct evidence we have of the nature of divination in Harry Potter. As the Pottermore article indicates, Trelawney is in some sense not a real Seer, and her gifts are very "diluted" compared to her ancestors. Could fortune-telling techniques be more useful to a full Seer? Could a full Seer make accurate predictions willfully? Since Trelawney is our main evidence, and her predictions are never conscious (or at least recognized), it really is hard to tell.
I suspect that Firenze has the right of it: Divination is an inexact art, which is not always right, and may often simply mislead those who practice it.