Brian Anderson writes:

Do I need a more powerful drill, or should I be using a different kind
of bit?

Drilling isn't gonna work.

What the world needs is a hollow carbide-tipped drill that fits over
Tapcons. Then you could just drill over the busted Tapcon, reaming out a
little ring of the surrounding masonry, and then replace with epoxy anchors
or fresh masonry fill. Kind of an anti-Tapcon drill.