For fixed-size integer types overflow is all but guaranteed to happen, yielding incorrect result. While for sum it is somewhat acceptable as the result cannot fit the type, this is not the case for average.
As the result of average is always within the scope of the source type, failing to produce it only signifies a bad implementation. Yes, one can solve this by type-casting, but do we really want to always have to be telling people that the correct spelling of the average function is cast(avg(cast(value as bigint))) as int), especially if this is so trivial to fix?
Additionally, the straightforward addition we use for floating point versions is not a good choice numerically for larger numbers of values. We should switch to a more stable version, e.g. iterative mean using avg = avg + (value - avg) / count.