This may be too much of a simplistic question: but is it correct to say that it simply doesn't make sense to compute averages/means of data that is fundamentally multimodal? That is, there is not one unique mean/variance?
There certainly is a mean and a variance of a multimodal distribution, and they are just as unique as for unimodal distributions.
Does it make sense to compute them? That is a more interesting question. I would say that it usually does not, but it might. For instance, if we were comparing the weight of Olympic team members (e.g. for purposes of loading a small airplane) then the mean weight might make sense, even though there would be modes for gymnasts (light) and field athletes (heavy) and so on. You might be able to come up with sensible interpretations of a mean in your application.
In general, though, any measure of central tendency (not just the mean) will be rather uninformative in a multimodal distribution.