yes, the coil might not be strong enough to pull the clutch hub in, possibly the spacers between the clutch and pulley are worn down or the clutch is worn down. A new clutch assembly (coil, pulley & clutch) will do it.
You could try and take the clutch off (1 bolt) and remove the spacer. Many time this will do the job for a while. Removing the spacer narrows the gap between the clutch and pulley and the coil does not need the same energy to pull the clutch in as it normally needs. It’s a free fix and only takes 30 minutes.
To test this you can use a feeler gauge and measure the air gap between the clutch and pulley. It should be between .020 and .030 and .30 is sort of pushing it, but check for specific parameters for your model and year as they can vary. If you do not have a feeler gauge use 2 business cards. If they do slide between the pulley and clutch the air gap is surely too large.
Edit: By the way, while you have the clutch off try to move the compressor shaft with your hands. If it moves very easy without resistance from the pistons inside, there is a good chance the shaft is broke and you need a new compressor. A broken shaft, obviously, would prevent the clutch from spinning even when engaged.