We need to clearly understand the difference between a closed source and open source non custodial wallet, also, i have seen some exchanges claiming that they are non custodial wallet and so on, developers have their own way of verifying for all these, and also in determining whether a wallet is closed source or open source code, because some of these wallet can lie on what they are not, except we know how to verify on their claims.
Centralized exchanges are service operators, if they don't lie and be dishonest about their business, they wouldn't be able to feed and wouldn't be able to pay their staffs.
I have noticed since there is a demand for this wallet centralized exchanges add to their wallet popularly called web3 wallets. They provide seed phrase but the wallet are attached to the exchange so people can trade like Dex but this apps don't have reviews, they are not open source, pure business like their exchange but you will see alot of users interact with them even keep large amounts of coins for trade and some for hold.
Just some months ago, Trust wallet extension was announced to be hacked and victims lost a total whort in millions. I don't know if the refunds were done by Binance after the announcement. It was later discovered that one of the Trust wallet developer rolled out and update which was responsible for the hack, if this were to be a open source, the chance of it happening will be less as the community will get to review the codes. Closed source or partial is a time bomb, don't use.