Upgrade to latest `fastapi`

[gctucker]

As a follow-up from #306, we should consider upgrading to the latest FastAPI (0.101 at the moment) to keep the codebase up-to-date. This also involves upgrading dependencies such as Pydantic, PyYAML and a few others.

It's not required for Early Access but would be good to have for the initial production deployment as the current FastAPI version 0.68.1 is from 2 years ago

[JenySadadia]

Addressed in #554.