Switch restore code path to use service-level handlers. Does a few
things:
* switches existing service-level functions to be methods on
service-level handlers
* update interfaces as necessary
* moves some logic from old controller-level restore function to either
the new handlers or the operation-level function
* removes old code
May be easiest to review by commit
---
#### Does this PR need a docs update or release note?
- [ ] ✅ Yes, it's included
- [ ] 🕐 Yes, but in a later PR
- [x] ⛔ No
#### Type of change
- [ ] 🌻 Feature
- [ ] 🐛 Bugfix
- [ ] 🗺️ Documentation
- [ ] 🤖 Supportability/Tests
- [ ] 💻 CI/Deployment
- [x] 🧹 Tech Debt/Cleanup
#### Issue(s)
* #4254
#### Test Plan
- [ ] 💪 Manual
- [x] ⚡ Unit test
- [x] 💚 E2E
d36636285a