There's two types of chain forks, soft and hard forks.

Soft froks are backwards compatible, it's a way of upgrading the protocol (see SegWit), and don't split the chain.

Hard forks are not backwards compatible and they split the chain into two separate chains.

Over the time, people has done both. The reason why hard forks, creating new "bitcoin-derived" shitcoins, haven't been so successful in the past is incentives, as you summed up. There are many angles as to why the incentives don't add up, but mainly is hashing.

If you keep the same algorithm, even if a lot of people was running nodes, one single big miner moving to the new chain could attack the network. If you change the algorithm, the hashrate would drop to almost nothing and anyone willing to spend the capital could gain the majority of the hashing power and attack the network. In both scenarios, the network is vulnerable, so it makes no sense for any serious player to pay divert their attention from Bitcoin and into the new chain.