Crypto trading takes place on exchanges, not the blockchain though?

In crypto, you have centralised (e.g. Coinbase) and decentralised exchanges (e.g. Uniswap). Decentralised exchanges operate onchain via smart contracts.

This article from a16z explains the mechanics of reordering transactions for profit (MEV): https://a16zcrypto.com/posts/article/mev-explained/