Improve catch_unhandled_exceptions

[gjoseph92]

• Catch multiple exceptions, not just the last one (not sure if this actually happens, but I imagined it could)
• Retain the full context message, which contains useful information such as the asyncio.Task, etc.
• Print full tracebacks when possible

Split out from #6329. Before, you'd just see an error like

test_client.UnhandledExceptions: Task was destroyed but it is pending!

which is very hard to track down. Now, you see

----------------------------- Captured stdout call -----------------------------
------ Unhandled exception 1/1: 'Task was destroyed but it is pending!' ------
{'message': 'Task was destroyed but it is pending!', 'task': <Task pending name='background-send-Client' coro=<BatchedSend._background_send() done, defined at /home/runner/work/distributed/distributed/distributed/batched.py:80> wait_for=<Future pending cb=[<TaskWakeupMethWrapper object at 0x7f2e00d00280>()]>>}

which is a little bit more useful to track down.

cc @graingert

• Tests added / passed
• Passes pre-commit run --all-files

[github-actions]

Unit Test Results

       15 files  ±0         15 suites  ±0   6h 57m 1s ⏱️ - 6m 47s
  2 792 tests ±0    2 712 ✔️  - 2    78 💤 ±0  1 ❌ +1  1 🔥 +1 
20 707 runs  +1  19 796 ✔️ ±0  908 💤  - 2  2 ❌ +2  1 🔥 +1 

For more details on these failures and errors, see this check.

Results for commit 7ba6e50. ± Comparison against base commit 63cdddd.

[gjoseph92]

Failures are distributed/tests/test_stress.py::test_stress_scatter_death which is known to be flaky #6305