A show can absolutely be too emotional, but what you're describing doesn't seem to be an example of that –– it's more a "triggering" thing.
For me, the idea of a show being too emotional would be sad/harrowing/depressing subject matter, sad ballad upon sad ballad, constant crying onstage, extended emotional breakdowns, and/or little levity. Let us feel the emotions ourselves. It's almost a copout for an actor to let the heightened bawling emotions overtake them for more than punctuation on a larger sequence.
Or something like "Peace for Mary Frances" (that awful Lois Smith/J. Smith Cameron play that Lila N directed at the New Group in 2018), in which Lois played a dying woman and it was relentlessly depressing and most of the characters were unlikeable or self-loathing.