Is there such a thing as a safe, clean, and family-friendly milk bar in or near Warsaw's old town?
Yes. There are a few around the centre. In the Old Town, there's a decent one (in the bit called the New Town) just north of the Barbican - you can't miss it. There are also some others spread around town.
That's good, however yes, there are often queues. Familijny on Nowy Swiat is ok, very authentic, as is Zlote Kurki on ul Marszalkowska.
I'd prefer not to get a stomach bug, food poisoning, or be attacked or harassed for money by a group of drunken bums.
That doesn't happen in Bary Mleczny. About the drunken bums, you will see plenty of these begging in Warsaw anyway. Occasionally standing outside a Bar Mleczny but rarely hassling people inside.
They do come into cafes though, even nice ones.