Improve Quart Tests

Our quart tests failed to detect that the Quart integration was completely broken. We should improve them so that they would have detected the failures in the below situation:

> It is very surprising that our test suite did not detect that this integration was completely broken. Perhaps, this is a limitation of the Quart `test_client` we use in testing. We should certainly add tests to make sure we do not break in this way again, but that can likely be done later, most important is to quickly fix the broken integration ASAP

_Originally posted by @szokeasaurusrex in https://github.com/getsentry/sentry-python/issues/3043#issuecomment-2093308286_