Is there anything that MUST be done on a multi-core CPU?
The question is: under what constraints?

There are certainly problems where, if we ask the question "can we solve this problem on hardware X in the given amount of time", the answer will be no.

But this is not a "future-proof" answer: things which in the past could not be done fast enough in a single core probably can be now, and we can't predict what future hardware will be capable of.

In terms of computability, we know that a single-tape Turing Machine is capable of computing all the same functions as a single or multi-core computer, so, runtime aside, there are no problems that a multi-core computer can solve that a single-core can't.

In terms of something like graphics, literally everything that is on the GPU could be done on the CPU... if you are willing to wait long enough.