Let's not confuse between exchange and wallet, an exchange doesn't have private key and they have control over your money, while wallet you have the private key and only person have private key can control the wallet. What you mentioned above is exchange and not a wallet (e.g. Bitcoin core, Electrum, Blue wallet).
Not all kind of wallet though, there's a sub-category for that made by businesses to make profit under the name of a "wallet", non-custodial and custodial wallet.
Custodial wallet works like an exchange, no private keys = not full control, but it still named as a "wallet".