@Kyle Houlahan People are being pretty hard on you, maybe for good reason, but based on your story, you had little understanding of the magnitude of the issue. I personally may have had the AC charged the day before closing, but who knows?
I've had an EXTREMELY similar situation happen to me on a house we flipped. The day after closing, the buyer communicated the AC wasn't working. As an investor, of course we want our buyer's to be happy. So yes, I did look into it for them. However, the final determination was that the buyer had ample time to perform inspections, including AC inspections. They did these inspections and did not find any concerns at that time. Additionally, like you, I had bought them a home warranty. Therefore, we didn't jump through any other hoops to get them taken care of. However, it has always bugged me a bit and my Lesson Learned from this experience was to go over-the-top, when selling to a retail client, to ensure everything is going to be to their liking after the sale is closed.