It's not really "sharpness," it's "depth" as in shallow or deep.
If you think about it, it's even more complicated. Because many (most?) sharpeners deliberate dull (round off) the tip of the edge, to make it more durable and less prone to being bent over, and so people don't notice as much the change in sharpness after sharpening. Rounding affects the depth of the hollow associated with a given ROH too. And it definitely affects effective sharpness.
The hollow creates a fairly specific "edge angle" - though if the blade is side honed (e.g., with a dovetail shape, widest at the bottom), that alters the edge angle that a given ROH generates. Edge angle is a big component of effective sharpness.
I prefer to repoint and polish the sharpening burrs into thin foil edges, which act very, very sharp - but are not very durable at all. But, while I worked at a rink, I sometimes gave up on foil edges, and dulled my edges instead, because I didn't always have time to put on blade guards, so the edges took a beating.
On top of that, the grit of the grinding wheel (or sharpening stone, if you do it by hand) has a profound effect on effective sharpness. Fine grit makes blades feel a lot sharper. I'm not certain I completely understand why.
But the o.p. may not have a lot of control over all these other factors, especially if they don't let him/her interact directly with the sharpener, to specify degree of rounding, deburring vs creating a foil edge, grit size, etc. So while ROH alone does not determine effective sharpness, it sounds like the only way the o.p. can affect effective sharpness may be to specify ROH.
One could argue that unless the prior blades were significantly thinner or thicker, the ROH on the prior blades might be a good starting point, because that is what he/she is used to - unless he/she wants to adjust ROH to make spin entry stronger (lower ROH) or to make the spinning friction less (higher ROH).