Even if dwarves are typically stable enough not to fall over when they stumble, that doesn't mean they'd escape unscathed if they happened to be carrying 10 ton boulders or bins full of toys made from similar boulders. Urist might drop what she was carrying onto her foot, shattering the bone through the pig tail sock.
I'd imagine tripping would mostly be a problem on rough floors and on tiles that are cluttered with items. Avoiding clutter in the main thoroughfares would be more important. Crowding could potentially be part of it, too, but I can't see a way for the player to deal with that problem, so that might be too unfair. Clear and smooth tiles should be pretty hard to trip on for all but the clumsiest dwarf, but smooth floors could get slippery when wet, so cleaning would be an important job for more than just appearance's sake.
All that said, tripping would have to be exceedingly rare for it not to get annoying fast. I'd think only very, very clumsy dwarves should even have a chance to trip.
The thing about workplace accidents, which I kind of like the idea of, is that they should be a function of skill. That would mean accidents would be more common in the early part of a fortress's existence, which would mean that not bringing along a skilled doctor in the beginning could doom a fortress to failure. That's kind of realistic, and kind of not. I don't think I'd like it much, personally, without a "suck it up and keep working" attitude among the dwarves -- at least the starting seven.
I know that the goal of the proposal is to give doctors work and experience, but I think there would have to be ways to mitigate the damage without doctors. Furnace operators might be less likely to burn themselves if they have heavy leather gloves and aprons, for example. Most jobs could have safety equipment like that. I guess I don't like the part of the idea that this is supposed to punish players who neglect the medical stuff. It's an open ended game, and I don't think there should be many "you're doing it wrong" elements like that. Besides, sometimes you're going to lack doctors because of ambushes, not because of any fault of your own, and then you could lose everything in an "injury spiral," where less and less skilled dwarves have to take on all the jobs while injuries put more and more dwarves to bed, while you wait for a new doctor to arrive in a migrant wave.
Okay, that's enough off topic rambling. I'm just going to get worse if I don't stop myself.