If you remove the jumpers and use the Main Inputs to bypass the integrated's preamp, it might well warm up the sound at an appreciable degree. Or you could get a really good tube phono preamp and come in through the AUX and still use the integrated's switching and tone controls, which might also warm things up. But neither would likely sound as good as going with separates, all tube.
I'm on board with the separates as well, but it is a step upward, both in cost and in sound quality. At least if you can get a tube preamp now, you always have the option of upgrading to a separate power amp at a later date and reap further benefits.
In the present situation, changing over to a tubed preamp is a mixed bag in concept though. At worst it could sound
different but not necessarily
better. Yet if the tubed preamp is better in quality than the Sansui, and the power amp section is up to the task, there could be a definite improvement. A tubed phono stage (run through a spare Aux input) is similarly baffling, but I think overall that most separate phono stages are better than what is built into many integrated amps and even preamps.
And finally, keep in mind that many of today's tubed preamps to
not have a phono stage. So, that is an added expense. It might be wise to start with a tubed phono stage, then a tube linestage later on...then work towards a separate power amp. And hang onto that Sansui for an alternate system!

In my case, I was looking hard at tubed phono stages but could not find any I cared for, and ended up with a Phonomena II+ that sounds phonomenal (sorry) through my tubed preamp.
A tube buffer might be worth considering.
I've gone that route and it just introduced more problems than it cured. It added a slight "tubeness" to the sound yet it also put a veil over the whole thing. (And now I'm stuck with one I have to sell... *sigh* ) I just see it as an additional circuit that mucks things up. On lesser preamps it is not noticed as much, but once you get a more resolving system, they stick out like a sore thumb IMHO.