Run cargo build and test built binary to ensure it works

[obi1kenobi]

I found it this by accident while testing something else with the locally-built binary for the newest version, so we got lucky. Otherwise, I think this would have made it through our entire release process and would have broken our users.

It's probably a good idea to add a CI check that does cargo build, then does a cd into the release directory and runs the built binary directly on some rustdoc JSON files (e.g. from the test suite). If we put it after the cargo test step then it probably will have minimal cost, since cargo test will already have done almost the entire build.

Originally posted by @obi1kenobi in #234 (comment)

#234 got lucky, and caught and fixed a critical issue that would otherwise have made it through our entire release process without getting caught. Let's amend our test setup to prevent it going forward.