I would submit the degree of influence is about 50/50, be it tube pre or tube amp in conjunction with its alter power mate.
I have personally heard however some tube preamps which do deliver that solid harmonic richness to the mix purely SS power trains can lack. The tube amp as well plays as significant a role though in providing the retro tube sound..
So, at times I think the choice to tube pre & SS amp is to tweak things, more so than to add purely sonic gains.. Things such as brightness, thin mids, dryness, analytical sounding etc.
A better intentional matching of a tube pre to an SS amp will indeed result in getting one closer to harmonic nirvana, but only going the whole route will insure that end.
Then too with the current state of tube amps, and preamps for that matter, the retro old school sound is only going to come from those topologies using single ended triodes. Its rrare in my opinion that PP tube amps can pull that sound off fully. Push Pull will benefit some facets of the sound and give greater flexibility due to their usually higher power ratings alone, but I think exceptions aside, the SET sound is indeed the SET sound and past it, all else is compromise.
As long as compromise isnt out of the question, mixing tube and solid situations is a wide open affair.
I suppose it comes down to just how much of a particular sound or influenced sound, you really want.
The problem as I see it with this controversy is this most audiophiles need two rigs. Not every stereo nut can afford to do severe justice to two rigs or do not have concomodate space for them hence we compromise or simply remain frustrated, always seeking out another angle on how to achieve IT.
TWO SYSTEMS CAN BE DONE IN ONE ROOM THOUGH using one pair of speakers with some thoughtful matching, and swapping speaker cables! It just costs more is all. and even then some smallish amount of concession is quite likely.
A conciliatory mind is better than one system which tries to serve two masters. and far less expensive to own and operate.