x/playground: simplify development and testing

[findleyr]

The playground development setup requires Docker, and as a result there is relatively little test coverage using go test (and therefore very little coverage in our default CI). CL 612456 is an example of a small change where a test would have been nice, but was not feasible due to friction and time constraints. Proof: in https://go.dev/cl/549015, I had actually started this fix for a related issue, and stalled because it was a low priority and I felt a test was warranted.

As a de-facto-but-superficial maintainer of the playground, I think this friction is holding back playground fixes and other improvements. From first principles, we should be able run the playground without Docker, and in doing so should be able to write end-to-end tests.

I think this is a good candidate for a friction fixit week.

[dmitshur]

Tracking issue #66002 is related.