Is it reasonable that a problem "solved" by a traditional ANN can also be solved by a CNN?
In theory yes in practice maybe.

What do I mean by that. There is universal approximation theorem for smooth functions regarding ANN, there is also theory backing this up for CNN variant. Another thing that backs this claim up is that you can replicate any ANN with CNN architecture, hence effectively backtracking to the original ANN.

Practice, you say this data can also be processed as images it can be processed but does it have the same amount of informativness as other data representation? Could be but it does not have to, think about doing regression with images of time series data, sure you can catch some general trends but to really catch all of it it will get expensive, really fast.