cmd/api: add JSON output support to the run.go program

[dmitshur]

This is a tracking issue for adding JSON output support to the cmd/api/run.go program, which is invoked by cmd/dist to implement the "API check" test. This is a subset of #37486, broken out into a smaller tracking issue.

Like in #56844, there are a few implementation approaches, need to investigate what works out better. One difference is that the cmd/api/run.go program is much smaller in size and scope compared to the test/run.go one.

CC @aclements, @golang/release.

[gopherbot]

Change https://go.dev/cl/453258 mentions this issue: cmd/api: rewrite as package test