I don't see why someone cannot dress nicely at work. I work at a vet's office..I know I'll get hairy, bloody and nasty..I do not wear white (although some do...brave souls). In other setting, people can dress nicely. Who is to say it's an establishment that is geared towards kids. I frequently eat at a restaurant that is not kid oriented..a quiet setting, nice plate settings..an adult atmosphere. Rarely are kids under 8 or so there, but once in awhile someone takes their young child there anyway and we all have to listen to noisy, messy kids rather than the piano player.
When I have a child, I'll expect that I myself will get dirty, puked on, etc once in awhile. But, my choice to have a child does not mean that others need to be grossed up by said child. (although I cannot imagine not wiping sauce off a child in the first place...I am fairly certain I wouldn't let my kid around sauce at all until they were old enough to eat properly...because ew) And I still wouldn't want other people's kid's nastiness on me.
I fail to see how a desire to stay clean and not have sauce plastered across ones person (and, likely missing out on the rest of their paid shift, as I cannot imagine a manager wanting their wait staff to walk around looking like they took place in a pasta massacre for the rest of the day) makes one sound "rich" "snobby" or "spoiled". Someone should not have to wear black, drab clothing on the off chance someone launches a kid at them.
I know parents are used to getting stickied up, etc. Just as I don't think twice about some dog hair on me when I walk out the door. But, not everyone is a parent or pet owner.