It's a combination of things rather than purely socialisation, it's genetics, personality type, some dogs just are not happy in the company of other dogs, it's also to cover themselves legally, perhaps the dog would be fine with others, carefully managed, but if it attacks another dog and the rehoming place hasn't stipulated this as being a potential problem, they could be liable.
On the Many Tears Facebook page, it sometimes says that it depends on the individual and to still enquire just in case, although the majority of their posts require a resident dog to 'show them the ropes' of living in a house. I think, hand on heart, that Bear thoroughly enjoyed being the only adored dog in the house, he went everywhere with us after Zak died, although has suddenly decided the pups aren't so bad after all. He's tolerated them but is now seeking to cwtch with them. Makes me happy.
I have friends who foster for DT, generally puppies or young dogs who wouldn't do well in kennels. Whilst fostered they are with 4 other dogs,, and usually blossom very quickly. Without exception DT have specifically looked for homes with other dogs for these youngsters, so the no other dogs ruling obviously doesn't cover everything. You may have already done so but I would suggest contacting them and asking them i they ever have dogs needing canine company.
Overall I agree that rescues apply unreasonable terms quite often. When I was home checking for a GSD rescue I pointed out to them that I wouldn't meet the criteria they were demanding eg not being left for more than 3 hours, never being kennelled etc.