You can counterbore with slot drills. You can also use what my Dad called a fishtail cutter…you take a spare drill of the right size, and grind the end square. Then you grind a little releif on each side so that you have cutting edges. You counterbore the hole first with the same size of drill, then the fishtail cutter will square off the bottom of the hole.
You can also use a D bit to do the squaring off.
Otherwise you can make a proper counterbore out of silver steel. The usual ones I make are made by drilling into the end of the rod the size you want for the pilot, and turn the outside to the desired counterbore size. Then set the rod up vertical and file cutting teeth around it with a triangular file. The trick is to make sure that you leave each tooth with the full cutting height, and yet don’t leave any of the original flat face of the bar. This is not actually as hard as it sounds. Then harden and temper the tool, light straw for cutting. Make a suitable pilot to fit the hole, loctite it in place if it wants to fall out. This style of cutter does not have anywhere much for the chips to go, and is only carbon steel not HSS, so take it quiet and back it out frequently if going deep. You can make another style that only has two cutting edges if you want to cut a bit faster, but they are harder to describe.
regards
John