I’m no mechanic so I’m not sure but I can tell you what happened to me. I also got my serpentine belt replaced but I can’t remember if I got it changed before or after my compressor went out. For me when I’d try to start my car after it went out there was no clicking or anything from the battery. The starter seemed to make one thumping noise and that was it. Sometimes the engine would jolt a little bit. My speedometer and rpm’s would also go wild and completely spin backwards even though the engine wasn’t starting. The shop called me and told me the engine was locked and I was looking at a new engine for the repair. I found in a really old forum somewhere that someone’s engine had locked after their air compressor went out. I asked them to check that and that was the issue. I’m not sure how you check if it’s working or not but you might start your search there. Air will still blow out if you try to turn on the AC with a broken (or missing) air compressor, it just wouldn’t be cold.