Or maybe Benjamin Moore Regal Select Matte? Is that a reasonable choice?

I used the Sherwin Williams showcase paint from Lowe’s in flat and have not had any issues cleaning it. I have small children also, one of which tries to draw on the walls any chance she gets.

" Unfortunately, stains are often difficult to remove from flat finishes. Unlike paints with higher sheen, a non-reflective, flat paint finish has a porous texture, which can trap dirt and result in burnishing when scrubbed or rubbed. However, there is one notable exception to this rule – Sherwin-Williams Duration Home®. Duration Home, which is available in matte, satin and semigloss sheens, is formulated with patented cross-linking technology that forms a smooth continuous paint film. This cross-linking technology not only locks in the colorants, it also provides superior resistance to staining, burnishing, and water-streaking or -spotting, and ultimately provides superior washability compared with competitive offerings. And it can be tinted to the deepest, most saturated colors your customers want. "

Can you fix the bad walls? Then you paint whatever finish you prefer...