"nginx does provide health_check in its commercial builds, but at this time I'm not too inclined to pay $1500/server/year for the pleasure."

But Varnish does it for free out of the box ! Quick example from the documentation (https://www.varnish-cache.org/trac/wiki/BackendPolling) :

backend b0 { .host = "phk.freebsd.dk"; .probe = { .url = "/probe.cgi"; .timeout = 34 ms; .interval = 1s; .window = 10; .threshold = 8; } }