Yes, the sentence indeed is ambiguous, but no, this is not an issue at all.
In fact, most sentences in most languages are in (partly) ambiguous. Our brains just automatically resolve most of these ambiguities from context and world knowledge before we notice they were even there.
Take, for example, the first sentence in your question:
Recently, I wrote a short paragraph in German about my son’s visit to the doctor and that he no longer needed a plaster cast for his broken leg.
Interestingly, this is the same phenomenon as you describe! "He" and "his" could refer to either the doctor or your son. But of course, you and every reader will automatically assume that it refers to the latter (and chances are that you didn't even notice the ambiguity until I pointed it out) because not only is it much more normal to go to a doctor if you have a broken leg compared to the doctor having a broken leg himself, but it's also more likely that you want to write about things that happened to your son, not the doctor.
It works the same way for "seinem Untergang"
(with a little twist because "vor" is ambiguous as well in this sentence - it may be used as a temporal preposition, translated as "before" - but I will skip that to keep it simple).
"Untergang" is usually used for complete destruction and doom (or as the literal nominalization of "untergehen"/"to sink (into water)", but that's obviously not the case here). As such, using it for the destruction of the plaster is actually some exaggeration - but I guess the same can be said about "ruin", so it's a fitting translation. The important point is that it is quite expected that the plaster is going to be destroyed - much more likely than your son expecting his "Untergang" very soon ("Untergang" isn't even expected to be used for persons at all). Thus, every native speaker will as per default assume that the former is meant. Only if you add some context to make the second interpretation more likely, there will even be a trace of doubt.