It does make sense. If the compressor's clutch does not engage as it should, little or no R-134 will be compressed. No heat exchange, no cood air.
Press the ac button on the dash and watch the top of the compressor, if the whole assemlby spins, then your compressor is working. If nothing spins but the pulley, then it could be the compressor's clutch, High/low pressure sensors on the line, or a fault in the AC control itself.
Good Luck