No, you don't need to cut them down.
People do it because it looks messy and you know how people are. We like to arrange and entertain ourselves "cleaning our gardens"
Natura obviously doesn't do that. And most plants survive in their
roots and woody stems, not their foliage. So, any cleaning does not make sense.
And cleaning makes the soil more prone to suffer from hard freezes. Mulching is very important to ensure survival of some plants.
During the cold winter, even if wet, generally molds and rotting is not a problem because the weather will be too cold. But you can do some cleaning in spring if necessary (like if too much mulch covers young seedlings).
Our projects:
in Portugal, sheltered terraces facing eastwards, high water table, uphill original forest of pines, oaks and chestnuts. 2000m2
in Iceland: converted flat lawn, compacted poor soil, cold, windy, humid climate, cold, short summer. 50m2