So I had this exact issue with my 05 2.4. Here is what I checked. A/c turned on in the cab, air blows from the appropriate setting (feet, defrost, front vents and gets harder on the recirc setting) ac pulley spins but clutch did not engage. Spent MANY hours watching YT videos and “diagnosing”.
My A/C clutch would not engage so performed the following:
- freely spun the compressor with engine off had good tension did not free spin
- checked 10a fuse with test light - all good
- checked relay with multimeter and swapped with horn relay - both good
- jumped low pressure valve harness -did nothing
- carefully power probes the clutch power coil at the compressor with engine off - engaged! But only when I gave it power
- jumped the relay at fuse box positions 87 and 30 iirc and clutch engaged with engine on and off
- swore at Jeep and took 24 hour break

- watched another video with an obscure reference that the improper pressure would not allow the system to engage
- connected recharge kit, started engine with A/C setting ON and decided to let it rip! SUCCESS! A/C clutch engaged on its own and cycled. I continued to add refrigerant following directions on the can, rotating as it dispenses. I let it run for few minutes and voila, air blowing cold!
Only cost me ~$27 for the kit at Walmart and it has the dye so if it does leak I can check with UV light. frustrating for me and I thought I was checking it all. I realized that the system had some pressure so I thought it was good but testing with engine running really gave me the true indicator that it was low. With the Jeep not running it gave me a false sense of charge due to the equalized pressure from the high side.
Hope that made sense one of my first ventures working on my, new to me Jeep.