ruff check fails on main in playground/backend_manager

Running `uv run ruff check .` on `main` fails with 11 errors in `playground/backend_manager`.

These were introduced with the playground merge in #24:

```console
$ uv run ruff check .
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/auth/jwt_handler.py:32:29
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/auth/otp_service.py:41:35
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/auth/otp_service.py:68:25
UP045 [*] Use `X | None` for type annotations
--> playground/backend_manager/app/services/bigquery_service.py:37:23
UP037 [*] Remove quotes from type annotation
--> playground/backend_manager/app/services/bigquery_service.py:37:32
UP045 [*] Use `X | None` for type annotations
--> playground/backend_manager/app/services/docker_service.py:36:23
UP037 [*] Remove quotes from type annotation
--> playground/backend_manager/app/services/docker_service.py:36:32
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/services/session_manager.py:52:36
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/services/session_manager.py:81:28
UP017 [*] Use `datetime.UTC` alias
--> playground/backend_manager/app/services/session_manager.py:167:28

Found 11 errors.
```

Steps to reproduce:
1. Check out `main`
2. Run `uv run ruff check .`

Start by running `uv run ruff check .` on `main` and inspect the reported lines in `playground/backend_manager/app/auth/jwt_handler.py`, `app/auth/otp_service.py`, `app/services/bigquery_service.py`, `app/services/docker_service.py`, and `app/services/session_manager.py`. Update only the reported lint issues, then rerun the command; done means Ruff reports no errors.

Written by the indexing model from the issue text.