You need to be back at your Mac to install a new profile to circumvent the whitelist, you can't do it from inside your phone.

Yes the whole of Safari can be white/blacklisted, although the way I have it set up if you're in an app with a webview it's still using Safari under the hood, and if that web view lets you browse to an arbitrary site then it can be circumventable that way. You can presumably lock it down further to prevent this, e.g. force the phone to use a specific VPN that you can then control to whitelist certain domains.

Every time I use Safari on iOS it seems somehow kneecapped by default, so maybe whitelisting the browser is no problem.