To me, what makes a first generation RH amp distinctive is that it's run in pentode mode with the feedback being of the plate-to-grid short loop variety. The original designer says that the 12AT7 is his personal favorite because it's synergy with that style of topology beat other driver choices. I tried a different driver before exactly following his schematic, and indeed, it did not sound as good as the solution he derived.
That being said, I've built and heard amps similar to the RH84 that did not use the 12AT7 and sounded wonderful. They deviated from the original RH topology in that they had output transformers with a UL tap, and were run in UL and triode mode. In those cases, the plate-to-grid feedback was still there, but much reduced. Pentode mode needs more feedback to be stable, UL and triode strapped do not. A very good example of an amp of this nature would be Kegger's KT-88 schematic, or the Compactron amp.
I find the RH schematic to be a good way to recycle output iron that lacks UL taps, especially smaller iron and smaller tubes. You get to run them in pentode mode for most power, and at single digit WPC sometimes you need all the help you can get. It's a stable and good sounding schematic. I tend to not screw around with it too much, because I've not found a change under those circumstances that made it any better.
If I have output iron that has UL taps, I would not build a RH amp as it would be a waste of those taps. If my output iron lacked taps but I still specifically wanted to run it in triode mode only, I'd also not use the RH schematic. (Note that triode and pentode modes are usually incompatible enough to provide easy switching, unlike triode and UL.)