No they don't gatekeep the information, your information is always available to you.

Maybe they don't offer the information in the specific way that you want, but that can also happen in web3(Propertery API, OAUTH2 which means you have to have static ip server for authentification,etc.).

You pay for the convenience of having your data aggregated in the specified format.

That aside, banks in the EU are forced to offer api services, as specified in the PSD2 standard.

I just don't see how doing everything in a web3 solves anything better than regulations.