The biggest problem I experienced with Screen Time is that when an app is outside of its limits, push notifications from it do not get delivered. So enabling the feature caused me to end up missing out on timely push notifications. Sometimes the notifications during the downtime got delivered later and sometimes they just never appeared at all.

This seems like a bug to me, or at best a very unintuitive side effect. I wish there were a way to still have push notifications delivered normally and only have Screen Time effect actual app usage.