Can excessive jarring or vibration damage mechanical watches?

1. Yes, some extreme shocks can harm mechanical watches.
2. No, I don't know how much. I'm sure there are some people out there who've done some sort of calculations. But I feel like what we see and hear are just estimates, speculation, anectodal bias, or other shoddy evidence (maybe check Watchuseek). Just like the Reddit pic you sent -- it shows beat up gears from a chronograph with a warning about golfing. Down the post, the OP says he's a watchmaker and that watch came in from someone who claimed that they wore it playing golf. But guess what? There was also rust on the parts. The OP said "oh, that was due to a bad seal, not related to shock". So how are we supposed to believe OP's opinion that the teeth were damaged due to shock but that other evidence of damage was due to a separate cause, that ALSO didn't cause or contribute to the teeth and gear damage? Shoddy arguments to me. And sports watches were made to be worn while doing sports--they were worn that way before the advent of modern quartz watches and G-Shocks. But of course they're probably not the ideal watch to wear, since we have quartz watches that can deal with shock better.
3. Yes, modern watches have some form of shock protection. How much really does depend on the watch.
4. Would I wear a mechanical watch while biking? I guess it depends. It depends on the quality of the watch (in some ways, a higher quality sports/dive watch should withhstand shock better, because it's going to have more shock absorption built in, but on the other hand, if your watch is expensive, are you really going to put it in a risky situation?), it depends on how bad your roads are, really, and it depends on the cost.
5. Let's just say this-- I bike commuted for about a year, and I almost always wore a mechanical watch. I went over potholes, and all of my watches survived. BUT, (a) I mostly wear dive-style watches, even to work, and (b) none of my watches cost more than $1000. So I'm willing to take those risks.

TL;dr-- yes, shocks might cause harm to mechanical watches. The amount of shock that can cause harm, and the risk of harm, are hard to calculate. Whether you decide to wear a mechanical watch while biking is a decision you have to make based upon your risk tolerance.

No shock-absorbtion system will be totally shock-proof. At least there will be fatigue.

So yes, the risk is real. As well as that of the strap breaking and the watch falling.

Nah. Soldiers have been shooting with mechanical watches for about 100 years. Your arm absorbs the bulk of the vibration. Here's an article about playing golf with a watch..

Also, I cringe every time I read "why risk it".

Any mechanical watch has the potential to be damaged by excess vibration. Up to you if you think the risks are worth it. I wear an automatic watch all the time but it's also not terribly expensive; so if it dies, I can just replace it.