For Indiana Jones it was snakes, for me it's earwigs.
Do they actually seem to be coming from the drain? Or could this be impression given since the drain is the lowest spot and they kind of end up there when they start to scurry?
Excluding the vents, the drain line should be a closed system from the tub to the sewer or septic tank so they should not be coming from there unless you have an unwanted opening somewhere which is letting them in and maybe letting water out. And speaking of moisture, how is your chalking around the tub? Earwigs don't need much in the way of an opening so thay may be coming from that way.