iPhone 8 keeps rebooting after Battery Replacement

Were you having charging issues before the repair? Or just battery life?

Even without definitive answers to those questions, two things that immediately come to mind. The new battery could be faulty. Even from reputable sources, sometimes this just happens. If you connect the original battery again, does it boot up normally?

Being that you’re getting the restore screen sometimes (but would still be applicable if you weren’t), this could also be damage to the logic board done during the repair. Pry damage is not uncommon around connectors. There are a few components immediately around the battery connector which communicate battery data—how charged the battery is, temperature, etc…. If these became dislodged it could cause behavior like this.

You may be able to tell with your eyes alone, but the components are quite small and it would be better inspected with some sort of magnification. You can see the two mosfets here between the battery and display connector, but those are not the only components that could be to blame, there are other smaller components under the sticker as well