To say that unchecked destruction of the environment won't necessarily destroy us is the same as saying that we'll be able to create closed-loop life support systems that can operate indefinitely without any outside organic input, and that we'll be able to do it fast enough to save ourselves. I'm highly dubious of that.
Something I'm pretty sure of though is that we can never destroy all food-producing life on earth by just eating everything till it's gone. Whatever's left will thrive and can be used to make food. Maybe not as tasty or as cheap, but we won't need a fully artificial life support system. Even in an extreme case of desertification, we can still grow plants indoors, and rain will still fall on our "deserts". And that's the very extreme! In reality, maybe the price of beef will go up a little as farmers have to spend more on feeding them.
You're confusing actually destroying the things we depend on with carelessly using them and not trying to make sure they survive. An obvious counterexample - we do our best to eat as many cows as we can, yet the species is thriving! Maybe fish farms will do that to desirable fish species too. We didn't even wipe out sperm whales despite whalers doing their best to find all they could and having no concern for conservation.
I agree there's a _risk_ of that happening though. We just don't know for sure. My concern is with people who claim to know for sure when actually they're taking a dishonest extreme position to make their political point stronger. Of course we should be careful not to destroy the environment because if we don't, the downside to being wrong is terrible. Somehow most of the people arguing about this seem to be victims of their own extremism and unable to distinguish risk from certainty.