Private Tabs Disabled in Safari

I have noticed a major issue with my iPhone. No matter what I try, I can't open a private tab in Safari. This is not good, as it means I have to manually clear my cookies (which is impractical) or disable them altogether (which breaks some sites).

As of today, I have an adult Apple ID, but my dad is the organiser of the Apple family.

I have tried disabling content restrictions in Screen Time, but it didn't do anything. Web Content is set to "unrestricted".

There is no switch or button in Settings > Safari.

I have no configuration profiles installed that could change this. All I have are a few root certificates, a bunch of fonts, and an APN cert.

Private mode is fully functional in third-party browsers, but they do not support extensions or userscripts and provide marginally worse fingerprinting protection.

Can anyone help?