If the rectifier is doing 100% of the job, it shouldn’t have much impact on sound. Sort of a big if in the real world. There can be a bit of sag when the amp gets pushed hard with rectifier tubes, which can be a desirable thing for guitar amps, but not so much for hi-fi. As long as in-rush current doesn’t pose a problem in a given circuit, solid state rectification can be a great way to accomplish that task.
There are always a bunch of variables in play, but IME output tubes have made the most significant differences, but I’ve also heard ample differences in the drive tubes, so don’t want to dismiss the value of exploring that avenue too (tends to be a cheaper approach!). If you have a tube preamp, those tubes can make a pretty audible difference too.
Because every circuit can use different values, and different brand components in support of a given circuit for the same tube type, every tube is likely to sound different if placed into a different amp. There’s also a lot subjective preference involved too, so ultimately, other’s opinions may give you some direction, but you still need to experiment and draw your own conclusions.
With that said, I’ve pretty much settled on RCA clear top 12AU7s as phase splitters, and Sylvania 12BH7a as a center voltage driver for my Dyna/VTA 70 amps. I’m currently running Tung-Sol 7581A outputs, but have also really enjoyed the Gold Lion KT66s. Prior to that I found the EH EL34s to be a great bang for the buck with good mids, and the Nesstone 6CA7 to have more solid bass. Have liked JJ ECC803S in my preamp, but like the Mullard CV4004 better. YMMV (more likely YMWV!)
My audio budget has historically been miniscule compared to some, but I was fortunate enough to re-tube and experiment quite a bit in the year or two prior to Russia invading the Ukraine. Tube prices have more than doubled in most cases since then. If you’ve got ample budget to play, have at it. If you need to be more miserly, I’d suggest doing preamp (if it applies) and driver tubes first.
