It's called a plug cutter.
image shamelessly stolen borrowed from Rockler
Cut your plugs from a scrap piece of whatever you're making your project out of.
Also, a screwdriver makes an excellent pry-bar for breaking the plug out of the board. I don't recall whether with- or cross-grain is the way to go, but it should take less than ...
I don't think there's anything you can do to completely prevent this on the wood front.
I'm tempted to recommend something like Cascamite or Aerolite as alternative adhesives, but I'm not at all certain these extremely strong and reliable glues would prevent the problem entirely. I don't believe it's about the glue, I suspect it's to do with a minor ...
It's been quite a while since I've used my plug cutters, but I always cut them longer (deeper) than needed, then just used a flat-blade screwdriver to pop them out. I don't recall whether prying with or across the grain worked best, but it should only take 2 tries to figure it out.
It may tear up the bottom a bit, but you put that in the hole and nobody ...
I found this little helpful article.
The important part was he had a thicker board, cut the plugs with his plug cutter (not going through), put a piece of tape across the top (over all the plugs) then on the band saw turned the blank with the plugs 90 degrees and cut off the bottom of the board, leaving all the plugs attached to the tape