This sort of breaks tor though, doesn't it? Tor works really well if you stay on the same circuit for a while since it reduces the chances you have a compromise

This sort of breaks tor though, doesn't it? Tor works really well if you stay on the same circuit for a while since it reduces the chances you have a compromised circuit. If you start getting recaptcha to block every exit node except those you control, you essentially have amplified your effective strength on the tor network.