Tbh I think this is poorly written law that have made you think this. The laws say when a whistle is needed. And then go onto say that a whistle is not needed for a dropped ball. So I'd argue that it isn't needed at all as dropped balls are excluded after the requirement following an injury.
A dropped ball is ceremonial enough to not require it. How many other restarts begin with the referee holding the ball?
I think that is an entirely untenable reading of the Laws here. The law is the same for DB as for GK, CK, or TI, so nothing in the laws is distinguishing whether a whistle is needed after an injury/sub/caution based on the ceremonial level of the restart.
Just as, following a sub before a throw-on, a whistle is not needed
for the throw in, but it is needed
because play is restarting after a sub, when play is stopped for an injury, a whistle is not needed
for the dropped ball, but it is needed
because play is restarting after a sub.
If this were not true, the requirement of a whistle "to restart play after it has been stopped for a . . . injury" is totally meaningless--as the DB is the way in which play is always restarted if it is stopped for an injury.
(This does point out some of the poor drafting, however--play is never actually stopped for a substitution; a substitution takes place during another stoppage. But I think, while poorly written, the intent is very clear: if play is restarting after an injury/caution/sub, the whistle is the formal way that a referee says "we're done with all that and about to start playing again." The fact there is a DB/TI/GK/CK doesn't change that.)
All that said, is it the end of the world if you fail to properly bow the whistle first? Not really--as you said, the DB is already pretty structured and sends the message. So it is technically incorrect to not blow the whistle, but if everyone is actually ready, hardly the worst mistake most of us will make in a game. (And I will confess that though I think this is a black and white requirement, I have caught myself forgetting to do so on occasion.)