As I see it, there are three main problems with this origin story:
1. The name predates the colonies. By a lot. The service in serviceberry comes from “sorbus” (see #2), which had been used to refer to a different plant (see #3) since at least the 1520s. People in Europe and other northern climates would’ve had the same problem of having to wait to bury the dead (and there were itinerant preachers long before Europeans – even the Vikings – ever laid eyes on the New World), so it is possible they used the plant as a similar cue for burying the dead. But…
2. The name is what etymologists call a loanword subsequently changed by folk etymology. Loanwords are words borrowed from other languages, and many of these sound similar to words that already exist in the language. So, for example, the Spanish word cucaracha sounds similar to “cock” + “roach” so we get cockroach. Sorbus, which comes from the Proto-Indo-European stem “sor” and means red to reddish-brown. sounds similar to service, so we get service.
3. The name initially applied to another species in a different genus. The red part of “sorbus” refers to the brilliant red fruits of rowan, or mountain ash, Sorbus aucparia (we have a native species, the American mountain-ash, or Sorbus americana). Rowan, by the way, also comes from a PIE word which means red. While the fruits of serviceberry can be dark purple, they are frequently a beautiful purplish red.
So there you have it, the true meaning of the name: red berry!