The crew read back and accepted the clearance properly. That is how 100% of clearances are received. I tried to get a full route readback out of CCS one day because our company had revised our original route because of a controller strike and after several attempts with ground control and tower ended up calling departure control on taxi out to see what radial they wanted me to intercept after departure to get out of there.
Everybody that flys down there knows how difficult it is to have a nonstandard conversation with ATC that isn't exactly what they are expecting to hear. Cleared to FL370 is it. Anything else in your flight plan is meaningless unless you have lost com. Lost com is fairly normal for short periods down there so changing altitudes would be very stupid.