I started doing this a few years ago. I've got a pretty good stash of most things. Still could use a few more.
As for drugs, the FDA keeps a running list of meds that are in short supply:
Drug shortage listI recently read where Adderal (ADHD med) is in a shortage and that there are roughly 50 million prescriptions for it currently. That's a lot of folks in for a rude awakening. There was a time not so long ago that there was a shortage of epinephrine and atropine, two commonly used in the ER meds. Some of these shortages are coming from problems getting workers in the pharma plants and some of them are because of supply chain issues. Keep in mind the majority of prescription drugs are imported from either India or China. (Something like 90% of antibiotics come from China.) Problems with the west coast ports or transportation issues can cause spot shortages to pop up unexpectedly.
As for bandages and other types of medical supplies, it's a good idea to have multiple kits of various supplies like that. Most things don't "go bad" very quickly as far as supplies go. Things like elastic wraps can deteriorate over time so it makes sense to replace them every few years. Most stuff in a good med kit is fairly inexpensive so it's a good idea to have a well-stocked supply.