Currently have RH on two bikes, Naches Pass on my Roberts and Barlow Pass on a 700c, they are stupidly expensive but what people say is true, assuming everything is set up right the one upgrade that will have the biggest overall effect on ride quality is tyres, and they are unreal, fast, light and extremely comfortable. In my opinion worth selling a few parts to finance, they elevate the quality of every ride significantly.
I also have some Conti race king 2.2s which are very good, although they are the most puncture prone tyres I ever had.
Oddly the closest I found to giving the feel of Rene Herse without the expense are Schwalbe Kojaks, the problem with them is they have zero tread, so on surfaces like wet grass or mud they are completely useless, they are fine on wet tarmac and anything dry though. Had them on most of my bikes at some point, amazingly low puncture rate. Obviously not as plush as RHs or as fast rolling, but comfy enough, about as light and cheap as chips. Got mine for £28 the pair a while back.
Edit
I forgot to mention these....
Sorry for the shitty pic, I may be wrong but I think these specific tyres have been discontinued, they came on my Fat Chance and have been on my Klein. They are a 1.6 slicks, narrower and faster than the Kojaks and about as comfortable, also zero punctures so far, probably jinxed them now though.
It looks like they moved to a more modern compound and renamed them Sport Contact II, I can't comment on those. But if I found these again I would buy them and would recommend them to anyone looking for a decent 26'' road tyre that is comfortable and quick.