Were this my home I'd be doing one of two things to eliminate the awkward drop under the threshold:
Attach a pressure-treated two-by lumber block to the face of the concrete. I'd cut it to fit the height of the concrete and the width of the outer door trim, and I'd install it with the milled edge upward to retain a nice corner radius. I'd use either concrete screws or expansion anchors and construction adhesive or silicone caulk. (You could even add a nosing of another 2x4 to thicken it further. Well screwed to the first, full-height board it'll be plenty solid.)
Build a full landing of either one or three standard stair treads deep. This can simply be a 2x4 box with 3/4" plywood on top and short posts to suit. I'd leave the plywood extending about 1 inch over the framing on the front and maybe the sides as a nosing.
By doing that you make it easier to fit any threshold you choose, you reduce the chance of future damage, and you eliminate the trip/snag hazard that exists now with that edge protruding.