instead of removing low area..........puddles in my concrete patio
Grinding down concrete in a wide area (enough area to fix your problem) takes special equipment (a large concrete surface grinder) and is usually not aesthetically pleasing. It will change the texture and appearance and often exposes the aggregate (think pebbles, rocks, gravel) which lies under the surface. It would work but you would have to hire it done, and live with the new look.

You might consider cutting narrow drain slots from the low area/depression out to the edge, at just enough depth to facilitate drainage. With a bit of creativity this can be done in a way that may not be altogether hideous. You would start shallow and go progress slightly deeper to provide just enough slope for water to drain down. It is a fairly common method, and you can do it yourself with a circular saw and masonry blade:



P.S.- trying to resurface/level it by troweling products on top of the existing slab usually results, over time, with you apologizing to yourself and anyone else who sees it.