»Private Access« means your photos are kept private FROM Safari, while still allowing you to select photos to upload in a specific instance. This is the way how Safari operates for a while. Nothing to do here.
Now to fix your problem of not being able to upload or download images using Safari:
- open
Settings> Screentime > Content & Privacy Restrictions
- toggle ON
> Media & Apple Music
- toggle to ALLOW
> Allow Changes
- toggle ON
> toggle PHOTOS OFF
.