I have a 100 year old house that at some point had front steps put in. For the last 35 years they have not moved at all. The problem is that they are not sloped so the rain runs off and away from the house. Instead, I get a puddle in the middle of the landing after every rain.

On the top landing it is concrete with the perimeter in brick.

Any way to a) grind off the top (so it can be painted with concrete paint) or b) some kind of cement added to the top that would hold up, even if it was at most 1/2 inch thick and went down to zero? Any other ideas welcome. Thanks

Epoxy surfacing compound. It sticks like crazy and can be troweled to 0.0 inches. This is what they use to resurface the steps on the NYC transit subway steps. It comes in a two part liquid and a bag of sand to the correct proportions.

You could do a concrete topping slab or a thin set paver that could be grouted in place, in either case you could get a slope to get rid of that puddle.