In those searches you probably saw my experiences. Actually had a property (JW Orlando) tell me that Teachers did NOT qualify for the state goverment rate, and would not honor it. Took it all the way to the GM, and they now specify that in the rate rules. Rather than accept rack, I ended up paying a M11 deal that had been offered for like $10 more.
Did not get clear answers for Marriott Corporate as to a chain policy.
Also (I think it may have been Socrates???) I remember someone on the board saying that some of the hotels reserve that rate for goverment employees traveling on business only.
I have used it on occasion since that incident, but it is rare to see it as an extremely good deal.
YMMV.