You answered your own question:
he supposedly got so many of his bones broken
It's anime world we are talking about, unnatural things are only explained by the lack of detail in the scene. Or maybe turning it into comedy of some sorts. If you have been watching anime frequently you should have seen it's a common thing with most anime.
For example: when Nami was sick there were a lot of details because writer wanted you to focus on that. Where as in Ussop's case he wanted you to focus on rebellion and were trying to hype the climax. I am pretty sure they never even gave it a second thought. They also wanted you to see how much people/characters were struggling to survive.
I can accept Zoro getting his chest cut and still being able to fight a man on a unicycle, but Usopp's case is too crazy.
This just proves how on point Oda is as per his illustrations, that he makes you believe and visualize the difference in characters. He wants you to see usopp weak and scared. But you need to understand as supporting character to luffy there will be times when he will stand up and defy odds. Oda isn't just depicting a seen every arc he draws, he picks up a theme, a moral, a life-lesson and a statement. To be delivered via his plot.
On a second thought, he was with chopper and he did put bandage on him. And chopper got skill bruh. That's really more than what anime characters get.
Mostly anime tend to show injured and wounded for less time. Or they just timeskip it/ fast forward it somehow.
In anime it is more about will than health. Specially if you consider the one piece world. There are numerous examples justifying it. Example: White Beard during marine ford, Shanks when he saved luffy, Luffy of course everytime when he is in a fight
Also, He is
- Warrior of the sea
- Sniper of the future pirate king
- ...To be continued. Sorry my hand got tired ;(