How can one cash out a large amount of bitcoins (or other crypto) without damaging the price too much?
There are Over The Counter (OTC) services that allow you to sell, usually to an exchange, that then discreetly sells the coins via market maker operations on their exchange.
You can sell lots of BTC in the OTC market - and you can also buy. See the following article:
https://bravenewcoin.com/news/how-to-buy-large-amounts-of-bitcoin-through-the-otc-market/