The connectivity-checking code would generally result in
NMManager:state going CONNECTING -> CONNECTED_GLOBAL -> CONNECTED_SITE
in the case where the connectivity check failed. The brief incorrect
CONNECTED_GLOBAL is bad, because clients might see it and do the wrong
thing.

Instead, when we are ready to switch from CONNECTING to CONNECTED_*,
do a connectivity check first, and switch to either CONNECTED_SITE or
CONNECTED_GLOBAL based on the result of that.