I gave up on linoleum for this–it just wouldn’t cooperate–I used a Staedtler Mastercarve block instead, which indeed does “cut like butter,” and my Speedball Linoleum Cutter Assortment. You could also use Speedball Speedy-Carve Block, which I’ve used on other projects to equally great effect.
I got the really fine lines I wanted, but the downside is I can’t print it with a press. I haven’t been able to get the coverage I want in my proofs since I have to use craft stamp pads. My printing inks require more pressure than the carving block can handle.
I did give it a go with embossing powder, and the effect is much mire to my liking.