Refactor test suite to make real requests to a real server (#131)

In the course of validating #125, we discovered that using the stdlib's
[`httptest.ResponseRecorder`][0] mechanism to drive the vast majority of
our unit tests led to some slight brokenness due to subtle differences
in the way those "simulated" requests are handled vs "real" requests to
a live HTTP server, as [explained in this comment][1].

That prompted me to do a big ass refactor of the entire test suite,
swapping httptest.ResponseRecorder for interacting with a live server
instance via [`httptest.Server`][2].

This should make the test suite more accurate and reliable in the long
run by ensuring that the vast majority of tests are making actual HTTP
requests and reading responses from the wire.

Note that updating these tests also uncovered a few minor bugs in
existing handler code, fixed in a separate commit for visibility.

P.S. I'm awfully sorry to anyone who tries to merge or rebase local test
changes after this refactor lands, that is goign to be a nightmare. If
you run into issues resolving conflicts, feel free to ping me and I can
try to help!

[0]: https://pkg.go.dev/net/http/httptest#ResponseRecorder
[1]: https://github.com/mccutchen/go-httpbin/pull/125#issuecomment-1596176645
[2]: https://pkg.go.dev/net/http/httptest#Server