Ok, so here are my thoughts (IMO, YMMV):
"Better served" is a relative term. Any tube amp, if set up incorrectly, can produce noise. Conversely, any tube amp, if set up CORRECTLY, can be dead quiet. I have Woo Audio WA6-SE Gen 2's (2 of them), which are OTL. They are dead quiet with the right rectifier and drivers in them. They work great with my Atriums (in fact, they work great with my entire collection (see my signature). Having said that, they will not drive high impedance (think Susvarna, et. al) very well at all. In contrast, my Woo Audio WA5-LE Gen 2 will drive anything (that I'm aware of) with no issues PROVIDED you use decent tubes in it. More than enough horsepower to drive the most demanding headphones out there.
"What amp would I suggest?" really depends on a few factors, most importantly will be budget. If you've got the spare change, I wouldn't hesitate in a heartbeat to get a 300B over an OTL design. There are some great 300B amps out there in a variety of budget ranges. If you can't tell, I'm a bit of a Woo fanboy, but there are plenty of other quality amps out there for you to try. I think that you'll find that the tubes themselves will offer you more diversity of sound across the board, while the amp manufacturer provides the "stable" platform on which to build your sound characteristics. Much like optioning a car, you still have to pick your preferred base model, then customize accordingly. FWIW, you can get a Woo Audio WA5 with speaker outputs if you want. Personally, I prefer amps with specific focus (headphone amps = headphones, speaker amps = speakers, etc.) Again, personal preference here.
"Other type of all-triode or SET amp..." For me, none that come to mind, although I do know there are some folks running 45 or 2A3 amps out there. I'm not particularly versed with those, so others might be able to provide more insight. (
@paradoxper might have some thoughts here).
Regarding price points - There's nothing wrong with picking up a nice used item from the Classifieds (or other used audio website) provided you do your due diligence. My Woo WA5-LE and one of my Woo WA6-SE's came from the used market, both have been working flawlessly. Good way to save some cash to put towards your soon-to-be expensive tube rolling habit...