On several occasions in the early Harry Potter books we're told that the purpose of a Bludger is to unseat riders from their brooms.

The Bludgers rocket around trying to knock riders off their brooms

_{^{Harry Potter and the Philosopher's Stone}}

and

Bludgers never concentrated on one player like this, it was their job to try and unseat as many people as possible

_{^{Harry Potter and the Chamber of Secrets}}

Why then, if this is considered a normal part of Quidditch gameplay, was everyone so shocked about the fall Harry has following the arrival of the Dementors in book three? Admittedly he falls 50 feet, but it seems like falls were a natural and normal part of the game whereas Dumbledore’s involvement and the reaction of his teammates and friends would suggest otherwise.

The team was worried Harry might have died from so high a fall.

In addition to the unusual circumstances that caused Harry to fall, he’d also fallen from high up, and his teammates thought he had died from it. They were likely more concerned in this case than in other cases of falling off a broom because of how serious it could have been.

“It was as though Harry’s memory was on fast forward. The lightning … the Grim … the Snitch … and the Dementors …

‘What happened?’ he said, sitting up so suddenly they all gasped.

‘You fell off,’ said Fred. ‘Must’ve been – what – fifty feet?’

‘We thought you’d died,’ said Alicia, who was shaking.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 9 (The Grim Defeat)}}

When Harry just broke his arm, they don’t all seem this worried - the severity of this particular fall and their belief he might have died likely made them more concerned than in a typical situation.

Also, they were likely also shocked at the Dementors being there.

In addition, it’s also likely that part of the reason that Harry’s teammates are so shocked when Harry falls off of his broom is the way it was caused. He didn’t fall off as a result of any of the typical Quidditch hazards like Bludgers - he fell off because Dementors were below him and he’d fainted.

“And then a horribly familiar wave of cold swept over him, inside him, just as he became aware of something moving on the pitch below …

Before he’d had time to think, Harry had taken his eyes off the Snitch and looked down.

At least a hundred Dementors, their hidden faces pointing up at him, were standing below.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 9 (The Grim Defeat)}}

Therefore, it seems likely that at least part of the reason they’re all so shocked, in addition to them thinking Harry might have died, is that Dementors got onto the Quidditch pitch.

It's a matter of interpretation.

When Quidditch players talk about someone being "knocked off their broom" or "unseated" they probably just mean that the rider has been dislodged from the correct position for broomstick riding. They might be upside down, or hanging from the brookstick by one hand, or otherwise temporarily taken out of the game: actual falls are apparently extremely uncommon.

Obviously, this is someone speculative. But I think it is the only reasonable way to make sense of the reaction to Harry's fall.

While the theoretical purpose of Bludgers may indeed be to knock players off their brooms, in practice all they really do at Hogwarts is cause minor injuries. This can be seen from the exchange between Harry and Wood shortly after the quote you cite in your question:

“Er — have the Bludgers ever killed anyone?” Harry asked, hoping he
sounded offhand.

“Never at Hogwarts. We’ve had a couple of broken jaws but nothing
worse than that. You don’t have to worry about the Quaffle or the
Bludgers —”

If the worst that ever happened at Hogwarts was a broken jaw, then it sounds like no one has ever actually been completely knocked off their broom and fallen like Harry did in Prisoner of Azkaban.

Thus, even though the students know intellectually that players can be knocked off their brooms and fall from great heights, they have never actually experienced such a situation at Hogwarts. Therefore, when it does actually happen to Harry it is a very big deal.