To have too much on one's plate and to have a lot on one's plate are well established idioms for having too many things to do, i.e. for something burdensome. But what one normally has on one's plate, in the literal sense, is food, and normally having a lot of food available to oneself is a good thing, not a burden. So, how did something that would stand for something desirable if it were used literally, end up as an idiom standing for something undesirable?
It is, of course, true that having too much food in front of oneself may be an unwelcome temptation to somebody who is dieting (or trying to), but I suspect that the origins of the idiom are in the times when such concerns were uncommon.
Given that the idiom is widely used in British English, it is safe to assume that it is unrelated to to step up to the plate, which is based on the baseball-related meaning of plate.
This question is not seeking an elaborate account of the history of this idiom, except in so far as that history illuminates the connection between the literal meaning of the words involved (having an abundance of food) and their meaning as an idiom (having onerous tasks to deal with).