I think that if the OP wants to increase her biosecurity measures, she should be consistent. There are many dangerous equine diseases out there, I’m not denying that. My point is that it makes no sense to single this one horse (and the previous owners) out. Good biosecurity is about having a consistent overall plan, not singling one horse and two people that have “cooties.”
The worries about various dangerous diseases are completely real. But horses that are coming and going from shows, clinics and group trail rides where they are mixed with horses from many different barns are at a much higher likelihood of bringing home diseases like flu, EHV, and strangles vs. a horse from a local barn that has poor management practices (unless it is a sales barn with many horses coming and going). Perhaps the OP should use this experience to re-think his/her overall biosecurity plan, because, again, singling one horse and two people out does not make sense.
The OP’s main concern seemed to be that this horse might be a strangles carrier. In that case, she should work with her vet to clear that item off the list, but truly any new horse coming to the barn could be a carrier. If the new horse IS a carrier, a short quarantine and being fussy with the people dropping the horse off isn’t going to prevent anything, because a carrier is not likely to either clear or get sick within a short period of time. Again, horses from the OP’s barn traveling to and from shows and events could easily be coming in contact with strangles carriers at any time.
Perhaps there is something I’m missing… if there is any evidence of current illness at the barn the new horse is coming from, the new horse should not set foot on the property.
For reference, some of the most famous and valuable racehorses in the world entertain many human visitors on a fairly up close basis. I can’t help but wonder if the restrictions the OP wants to place on the people in question are based more on an emotional response to his/her opinions on their horse management vs. actual risk.