Honda CRV (2001) AC System - Should dealer have told me or fixed problem?

My wife and I have a 2001 Honda CR-V with approximately 91,600 miles on it. Standard transmission. Air conditioning.
About three weeks ago, she reported that she suddenly heard a loud noise and the AC went out. No engine light, but the noise from the engine compartment was loud and she was concerned about the safe operation of the vehicle.
We took it to a Honda dealer. Their diagnosis: Just the AC system, which needs to be replaced. Their estimated cost of replacement was $3800. I deferred on paying that price for replacement of the AC system. They stated that the vehicle was ok to operate, but we just wouldn’t have air. We paid a fair amount for their evaluation of the problem. (I also asked them to look at the clutch, which they said was ok.)
The noise continued from the hood of the car.
In the meantime I got a quote over the phone from another mechanic that replacement of the AC system could be done at significantly lower cost.
Just before we were going to take the CRV to the mechanic, I get a call from my wife to inform me that she’s stranded. The CRV won’t start and run.
We called AAA. Their guy comes out and simply cuts a belt to the AC compressor clutch. Problem solved and noise eliminated. The CRV is now with the other mechanic.

Why didn’t the dealer tell us about the problem that would be caused by the belt? Why didn’t they just cut the belt in the first place?

Cut a belt? You don’t mean cut a wire do you?

Obviously the compressor locked up. You’re lucky there was a separate belt for it. No way the mechanic would have known that the thing would lock up and not free wheel but maybe someone turned the air or defrost on and the thing destroyed itself.

Regarding Keith’s comment: It was definitely a belt.