Reconsider code coverage approach

We can close this issue when we have documented why we should not do these things **or** we have implemented them.

With the code coverage build moving to GCB (Google Cloud Build, see #3880) it may be time to reconsider how we run this build. We use `cargo tarpaulin` because it was easy to set up, and was not too slow when compared to `cargo llvm-cov`. That may have changed, as the code size has grown.

We also exclude all the integration tests because we could not run them in GHA (GitHub Actions). That is no longer true.

In summary:

- [x] Use `cargo llvm-cov` to run coverage builds.
- [ ] Enable as many integration tests as possible as part of the coverage builds.
- [ ] Trim the list of directories excluded from code coverage analysis.