I'm not really sure I follow your complaint, so I wonder if there's a point of misunderstanding. The tests written for DOM APIs certainly require running javascript to execute the test — that's a given — but most of the rendering tests are reftests. These consist of two files, one using the feature under test and one avoiding that feature [1]. The test passes when the two renderings are identical. That does make these tests difficult to run "by hand", but in terms of implementation it's possible to automate in anything that's capable of rendering to an image. Reftests are preferred over comparing rendering to a fixed image because fixed images tend to be invalidated due to unrelated/permitted changes in the rendering (e.g. a different font or different antialiasing choices). These will typically apply to both test and reference and so are handled automatically.
Reftests obviously aren't suitable to bootstrap getting very basic things working, so there are some manual tests; I suppose one could hope to avoid that by inventing a special kind of output inspection automation just for those basic tests, but it's hard to justify.
IN general, however, the "the ability to point my software at some .html files, get some failures, add more implementation, get less failures, and repeat" is exactly what wpt offers. There is some work required to stand up the necessary infrastructure to run the tests automatically, but it's common for new implementations of the platform to make use of web-platform-tests (e.g. Servo and Flow have both made use of wpt).
In terms of coverage it's very difficult to demonstrate that all the requirements of a spec, and all its interactions with other specs, are completely covered. There have been various proposals for adding test metadata to try to measure coverage, but these have largely proved impractical. Nevertheless some CSS specs have links from the spec to the relevant test cases. And if you find cases that aren't covered by exisitng tests it would be great to get a PR to add new tests [2]
[1] https://web-platform-tests.org/writing-tests/reftests.html [2] https://github.com/web-platform-tests/wpt