How to reinstall iron stair railing in concrete that rusted out at the base?
Use the core drill you mentioned before, make sure it is deep enough, say 4" to get to the bottom of the iron stub. That is the standard mounting depth.

To keep the drill in place, drill a hole in a piece of 3/4" plywood big enough (1'X4'??)to lay on the stair you need to drill with the hole in the place that the old post is located. Stand on this while you use the drill to cut the hole. Get another body on it too if you feel you need to, but if you can maintain good balance you will be fine. drill it down about 1/2" and set aside the plywood and finish the depth of the hole. you can always chip a little of the concrete out, to go deeper, but you will find that a well placed tap with a hammer and chisel or steel pin to the side will release the plug that you drilled.

The plywood will stay in place very well with the pressure of you standing on it, it does not take a whole lot of pressure to keep it from "dancing".

If you can use a water fed core drill.