I didn't find anything that I couldn't cut effortlessly, with my bevels polished to a 2000+ grit shine. It's more about the purity and integrity of the edge's apex. If that's good, it really doesn't matter to what degree of polish one finishes. The most common mistake made, I'd bet, when 'polishing' the bevels, is getting a little over-zealous and rounding over the edge itself. It takes longer, and requires much more effort and focus, to correctly finish an edge to a higher polish. And it's therefore easy to make more mistakes along the way, and blame the poor cutting on the 'polished edge'. Keep the apex as pure and tight as possible, along the entire length of the cutting edge, and the results will speak for themselves. BTW, this also applies to a toothy edge. If the apex itself is in good shape (geometrically), and all burrs & wires have been eliminated, a coarser, toothy edge can do many of the same things normally attributed to polished edges (like push-cutting).
EDIT:
I've started to view a 'good edge' as it might be seen in cross-section. More so, if viewed as thousands of paper-thin, cross-sectional 'slices' of the blade itself, stacked end-to-end, from heel to tip, to form the 'whole blade'. Imagine a push cut into the edge of a sheet of paper. In order to do that effectively, only the very narrow segment of the edge in direct contact with the paper needs to be 'sharp'. If adjacent cross-sectional 'slices' of the segmented edge aren't quite so sharp, the blade might still be able to push-cut. But a draw or slicing cut would be dimished, because not all segments of the edge are equally sharp. If, on the other hand, each individual cross-sectional 'slice' of the segmented edge is equally able to effortlessly push-cut on it's own, then a slicing cut, utilizing all of those segments in a seamless sequence, would be automatic. This might seem a somewhat convoluted way to view it, but I've found it useful. It's what I picture in mind, when I emphasize a pure apex along the entire length of cutting edge, from heel to tip.