It is possible to trade directly with your non-custodian wallet with some exchange like coinbase. And if you trade on dex, you will also be in charge of your coin. But to trade on centralized exchange, you will have to trust that exchange! When you keep your coin on centralized exchange for days or one or two weeks, you shouldn't have to worry about hack if those exchange have track record of safety of customers funds!
I think most daytraders store or leave their coins in the exchange itself.
Because getting out will incur fees, so if you are active trader, it is not practical to get in and out your funds frequently.
However, if you are storing it to the exchange, it is better to choose the exchange which has very good reputation.
And at least your funds are protected just like in Binance, which has insurance for its users.
Coinbase has also this FDIC-insurance up to $250k if you are keeping cash with them.