Hey there Julianna Kahn,
Thanks for reaching out to Apple Support Communities. I understand you have some concerns about using Screen Time on iOS 13. I’ll be happy to help with this.
I just checked on one of my devices running iOS 12 and Safari is not showing up under Always Allow there as well. This looks to be expected and there is an option to restrict Safari under Content & Privacy Restrictions > Allowed Apps. You can read more about this here:
Use Screen Time on your iPhone, iPad, or iPod touch
As for the Ignore Limit not being seen in Safari, does this happen in both landscape and portrait mode? There has been a recent update released, iOS 13.1, that includes many fixes. Back up your iPhone, install the update, and see if this helps.
How to back up your iPhone, iPad, and iPod touch
Update your iPhone, iPad, or iPod touch
Best wishes.