How did having too much on one's plate become an idiom for burdens?

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.

This is not the view that the idiom expresses. It has nothing to do with "having a lot of food." It implies that the amount of food on this occasion is really excessive but nevertheless, it must be eaten. It is a metaphor. If you have ever had to eat more than you wanted to eat, or been given a plate piled high with food when you are not particularly hungry, you will understand.

The idiom is relatively recent. The OED has:

27c. colloquial to have a lot (also enough, plenty, etc.) on one's plate and variants; (also) to have a full plate, to have one's plate full (up) (with): to have a lot of things occupying one's time or energy; to have a lot to do.

In quot.1911 in an extended metaphor.

1911 E. Wharton Ethan Frome 15 Sickness and trouble: that's what Ethan's had his plate full up with, ever since the very first helping.

1928 Daily Express 4 July 9/2 Can you tell me how many times in all she has forbidden you the house? -- No, sir. Half a dozen times? — It might have been. I cannot say. I have a lot on my plate... -- Mr. Justice Horridge: A lot on your plate! What do you mean? Elton Pace: A lot of worry, my lord.