Please help me understand the economics behind Proof of Stake

I'd like to discuss the economics behind the supply of PoS coins. In PoW, the economics is understandable – it's almost exactly like gold mining.

Coins are mined slowly in a steady stream and the supply increases slowly, which corresponds with a high demand (exactly like gold). With a capped max supply, this means that prices of PoW coins will generally increase if there's a demand for it.

However, in the case of PoS, coins are start by "selling pre-mined coins or they launch with the Proof of Work algorithm and later switch over to Proof of Stake" (taken from Binance Academy). While I understand how the economics for switching over from PoW to PoS (like ETH), I am quite skeptical of the part where pre-mined coins are sold.

Premining is associated with initial coin offerings (ICOs) as a way to reward founders, developers, or early investors into the project. Because premining effectively dilutes the outstanding stock of tokens, large premines are often frowned upon in the crypto community. (from investopedia)

With premining, there is a huge release of supply of coins at the beginning. This is in comparison to PoW coins, where coins are slowly mined one by one. When there is a significant release of supply, there needs to be a high demand to support it in order for prices to rise.

This is why quite a number of PoS coins' prices are pumped sky high on ICO and then they fall drastically. In the case of PoW coins with little or no pre-mine, prices have steadily increased (BTC, SC, LTC, DGB (small pre-mine) even DOGE lol).

Some PoS coins have introduced the concept of coin burning, which I can get behind (supply is reduced, price increases). I also would like to know, when coins are staked in a PoS model, what happens to the coin if the forger validates an incorrect transaction? I know the forger loses his/her stake, but what happens to the staked coin? Is it burned permanently? Is it distributed to the network?

Please help me understand the economics behind PoS. Thank you.