How can one cash out a large amount of bitcoins (or other crypto) without damaging the price too much?
If you're going to cash out a significant amount, you're going to definitely end up damaging the price. The only way I could see you not damaging the price is if you sell directly from peer to peer. If, let's say you have 5 friends who completely want to buy everything from you, then you could sell to them, and the bitcoin is still in someone's hands. People won't see a huge sell order and they won't try to pull out of the market asap.
If you're going to use an exchange, however, you should be prepped for the fees too. I'm sure that there may be some dangers (especially stuff like exchanges ghosting your account). Be careful with how you cash out!