When done well pithy musical satire can be extremely amusing. To be successful the protagonist of this genre needs to be clever, a bit of a wordsmith. A classically trained voice isn't necessary. It would probably take away from the performance, forcing one to pay more attention to the voice rather than the verbal gymnastics. For me Tom Lehrer is one of those individuals who succeeded in this musical niche. His, "Poisoning Pigeons in the Park." does I think show him close to the apogee of what was for him a secondary profession. His performances, now long ceased although he is still alive, showed him to have a quick witted and laconic style.
My autumnal peregrinations provided me with the opportunity of viewing the current musical presentation of the female triumvirate known as, "Fascinating Aida." I expect that they would see themselves as modern day protagonists of Lehrer's genre. Yes, certain of their songs proved to be quite funny but I don't know whether it would be fair or correct to describe them as satirical. On occasions somewhat risqué, at times bawdy and rather strident. Probably a good night out for a hen party, but I don't think that they are challenging Mr. Lehrer.