TubeNet

Is it possible to construct any kind of valved brass instrument using something besides rotary/piston valves? If so, what would the benefits be? Perhaps even Hagmann valves if you can fit them in there.

There has been a lot of work to make the Axial Flow valve (also known as Thayer) be a better valve, thanks to the fact that it has become so popular with players. There is a new version of this valve, called the Infinity Valve, that can be purchased here:

https://instrumentinnovations.com

This valve has sealed bearings on both ends of the valve, and it's amazing how little wear there is on the bearings in this. I've seen this valve hooked up to a cordless drill, drill turned on for 30 minutes, and there is no heat at all in the valve casing. Truly a fantastic product, only available currently on the Bach Trombones with the AF at the end of the model number, as far as I know.

I have seen a Gronitz tuba with Lätzsch or Hagmann valves (on their old website) and a Rudolf Meinl 5/4 BBb with Lätzsch or Hagmann valves.
So it can be done ...
Personally, I don't know why one would want to make the Rudi 5/4 BBb more freeblowing than it already is, I like a little bit of resistance personally...