Which has a greater influence on sound quality, the amp or the preamp?

Through the Sixties through the two thousands, I had. separates ( with one exception). Everything I changed, changed the sound. I bought a Hegel integrated a couple of years ago and it’s great, but early this year, I went back to separates and I’ve been having a blast rolling tubes again.  And yes, tubes make a difference as well.

When I added the PassLabs XA25 to my system many years ago now, it made the single biggest impact because it took over from an average AV pre/pro combo. It took control in a good way of the speakers I had at the time, PMC Twenty26's. They really shined in full bass to great mids, and highs where ample decay prevailed. Later on, adding the XP20 did not make as much difference at all, and I looked to sell it. There are obviously a number of dimensions involved: power and control versus tonality and timbre, and many more, Each component impacts these more or less. Generalisations are nearly impossible in this domain. That's why trying and listening for synergy are the most helpful pathway-makers.

....and all 'n all, it's just synergy, y'all.....

That, and your preferred POV.... ;)

They both offer different sound quality and sound characteristics.  

"Is it the inside dirt or the outside dirt that matters most?" assumes a nonexistent hierarchy.  It's the side of the glass with the most dirt tempered by whether fingerprints or water spots bother you more tempered by what floor you are on and how tall is your ladder.