You hit on some common misconceptions, but you also hit on some truths. For instance, you compare the human body to inanimate objects with respect to damage, but you also accurately point out that
Obviously, living organisms are not the same as inanimate objects
So, what's the difference?
The repair process
Man-made objects tend to wear out faster with more use.
Imagine you have an axe. The axe undergoes wear and tear and becomes dull and damaged. Dullness can be worked around by grinding the blade even more, but eventually, the blade will get cracked or even break. And that's not something that can be trivially repaired.
But imagine if the blade got a crack, and all you had to do was throw some iron ore on it, and let it rest for a couple days, and come back to a blade that was even stronger than before. The crack is not only repaired, but that's now the strongest part of the blade. It was rebuilt more dense than before.
That sounds farfetched for an axe blade, but that's how our body copes with exercise. Exercise is the act of damaging your tissue - be it muscle, tendons, or what have you - then giving your body time to eat and rest while it repairs itself under the assumption that "we'll be doing this exercise more, so let's prepare for that". And that's where we get our longevity too. With wear and tear, your muscles get stronger, your tendons get more durable, your lung capacity increases, and more.
More acute damage
You hear about people developing an injury because of their workout
So when you go to the gym and lift some weights, you damage the muscle tissue, which is then repaired stronger. What about if you actually injure yourself during training?
Anyone who works out for any non-trivial period of time will need to spend some time learning how to do each exercise in a way that minimizes or outright prevents unexpected injuries like falling, breaking bones, dislocating a joint etc. But at the end of the day, it only takes a momentary lapse in judgement to sustain an injury like that. So every individual has to make that decision for themselves. But it's not like you go to the gym and immediately start doing dangerous things. One thing we preach harder than anything else is "start light, and use proper form". Proper form is what prevents injuries, and if you're unable to do an exercise with proper form, then you're using weights that are too heavy. Exercising with proper form is and feels safe. At any and all points, you should feel like you're controlling the weight, and at no point should you feel like you're being yanked around by the weight, or having to catch it falling from a height. That would make you more injury-prone.
And I've heard about exercise releasing more "free radicals" into your body which are supposedly bad for you.
Pretty bad. It causes cancer! But then again, we subject ourselves to so many things that cause cancer. So why is it that so few of us actually get cancer? Because our body has ways of dealing with things that might develop in that direction.
I use the term "cause" pretty loosely here in an attempt to mimic alarmists. Like a lot of things that "cause" cancer, free radicals are an important part of our bodily functions. I'm no expert on free radicals, so I'm not going to spend too much time on it, but I found this Q&A by Dr. Harvey B. Simon of Harvard, who is probably more suited to explain it.
As well as "increasing inflammation" and other bad-sounding things.
I'm not sure this is a concern, per se. An inflammation is your body's response to something. It's how it deals with certain types of injuries or infections. It would be like being worried about how blood is leaking from your skin, rather than being worried about the knife stuck in your arm.
There are diseases that cause inflammations where it's not needed, but if you have problems with inflammations after exercising, it's likely you've been injured in some minor way, and we're back to preaching proper form.
But if exercise causes these problems for people, why do we believe that it is somehow healthier for you? Is it because studies indicate that exercise helps avoid things like heart disease and cancer? Do those benefits come at the lesser cost of those downsides to exercise? So we might say, "Years of jogging gave me a bad knee but it helped me avoid heart disease, so it's a net win."
Yeah, I think you hit the nail on the head on this one. I would argue that most runners do NOT suffer from bad knees though, so the vast majority simply enjoy the improved cardiac functions. Most of us do NOT drop weights on our heads, so we simply enjoy having stronger muscles compared to before we started training.
There are pains we go into the gym expecting, though. Like delayed-onset muscle soreness and "the burn". But these are the harmless, temporary pains that are your muscles' way of saying "phew, that was exhausting, I need a break". And then that, combined with a proper diet and plenty of rest, is what gives us that repair that separates us from inanimate objects.
But I'm wondering if somebody can point to why it's true. Rather than just taking it in good faith.
I just had to quote this one too. In the fitness sphere, it's very easy to find some guru on youtube who says "do this exercise, and don't do that one", and millions will just take it for granted that this must be great advice, because this guru has six-pack abs. Meanwhile, if you make sure to learn WHY an exercise is good or bad, not only are you equipping yourself to work out smarter, you're also learning anatomy.
Working out is an endless learning process about how the body works. Hell, I'm not an expert by any means, and probably never will be.
EDIT: @Jester, in a comment on the question, points out a truly good point that I overlooked. The body's "use it or lose it" response to inactivity. You're not using your muscles much? Then your body won't spend energy preserving them either, and they atrophy.