I think that, much like our beloved NHS, our railways have become sausage machines intent on moving as many people for as little cost as possible.
Passengers on trains with Tandems are as inconvenient to this as patients with (say) unusual dietary needs, or three legs.
.., and like the NHS, the exact basis for the rules are lost somewhere in the mists of time.
I'm guessing, that like getting bikes over the Dartford crossing, at some point in the past, when bikes as a form of transport were more common, and more acceptable, the government of that time put some legal requirement into the legislation that forced companies that run train services to carry bicycles.
Over time, they seem to have managed to bend these rules to a degree, to stop carrying bikes when the trains are busy, and only provide facilities for a minimum number.
I wonder why tandems and tricycles seemed to get left out of that legislation? Presumably they have always been relatively rare, so someone just forgot about them or ignored them.
As has been said, children are one way to make people more likely to help, and recumbent trikes also often succeed in getting where strictly speaking the rules don't allow them, because people assume that the users are disabled (and sometimes they are, but not always).
In some respects it would be nice if the rules were clarified and organised across train companies, but there's always a risk that this may just allow them to universally refuse to carry tandems.