healthcheck: answer 503 when no root server gave a usable reply in the last round of DNS checks (closes #302)

The health check answered ok while every DNS lookup failed, as on a
network that intercepts DNS queries. A lookup whose root servers give no
usable reply now fails with ErrNoRootServerAnswered. At the end of each
round of DNS checks that shutdown did not cut short, the watcher records
whether every lookup failed with it or with ErrIntercepted; the health
check then gives status error, and /.well-known/healthcheck and /health
answer 503. Until the first round finishes it stays ok.

The health check now depends on the watcher, so fx starts the watcher
before the HTTP server and stops the server first. NewWithRootServers
lets tests start a resolver's walks at real servers that are not root
servers.

Model: opus-5-5

TODO.md

@@ -19,6 +19,10 @@ milestone: https://git.eeqj.de/sneak/dnswatcher/milestone/7
 
 # Completed Steps
 
+- 2026-10-18: the health check answers 503 with `status` `error` when every
+  lookup in the most recent round of DNS checks failed because no root server
+  gave a usable reply, as on a network that intercepts DNS queries (closes
+  #302).
 - 2026-10-16: the first reply marked "recursion available" from a root server's
   address, a refusal included, stops the lookup with "this network intercepts
   DNS queries", so on a network that sends DNS to a resolver dnswatcher and its