I finally got the screen up but shattered it in the process. I was replacing it anyway but if the screen is not what you’re replacing, take EXTEME caution. The suction cup and spudger were not working for me to get it started. Thanks to the comments, I used a razor blade and a heating pad and went back and forth, back and forth, between heat and pressure from the blade. Once you get a gap, the spudger works great and the screen comes up easy.

Thanks for the tip on the new playing card and rubbing some isopropyl alcohol on the corner of the card! I also peeled back the metal plate on the broken screen to get that bracket out of the way for better access to the cable. After applying heat and slowly and steadily working up the cable with the card, the home button popped out in proper working order.