Most interior trim paints are very durable and hold up to repeated washings. Not sure how your kids get the baseboards dirty...they must work at it! Meantime, from a sanitary point of view, I kind of like knowing when stuff is grubby...let's me know when to clean. Hiding dirt is OK for awhile..until flu season or something like that.

In terms of decorating..I am going the same way. All stain or all paint, at least in a single room (doors can stain no matter what but they stand alone). A single color scheme on trim unifies the look and keeps things simple.

The only mix I really like is the combination of white trim throughout the house, but with solid wood stained doors and bannisters. But I think that mixing wood trim on some areas (baseboards, etc.) and white trim on others (windows) could look a bit chaotic.

Doing all stain in kitchen and family room us ok if kitchen is adjoined with with the family room and do the rat of the house in paint or do all paint and stained doors . Front door can be stained
Mantles or built in book cases look nice stained