You can nitpick anything and claim confusion about just about any topic. Why is 2+2=4?
Because, if you understand the meaning of the symbols "2", "+", "=", and "4," then the logical relationship between the parts as presented follows analytically from the meanings of those symbols. Or, if you prefer: "When you have two cows, and you receive two further cows from your wife, you will find you have four cows. Moreover, this relationship will hold even if you replace 'cows' with some other countable object."
The point is, the game
doesn't actually say something that is a perfectly valid question to ask. It
isn't like "2+2=4," because the definition of "combat" and "initiative"
does not actually include who determines when combat should occur, or how, at least not in the quoted text. Perhaps it does say so in other text we could quote! I'm hardly an expert on 5e, having never run it (and with negative desire to begin doing so.) But it would, at the very least, be a bit strange. It would be like defining everything
except the equals sign, and presuming people will just
know what it means from context. Some people will.
Some won't.
I would know. I have previously worked--for multiple years--with adult learners who have never had
any mathematics education whatsoever, and are thus being exposed to the foundations of mathematics for the first time
ever. Such folks need things explained carefully, but also with great kindness and respect, otherwise they will disengage, feeling overwhelmed or ashamed or the like. Their curriculum was
very carefully structured, to the point that an actual typo in one of their books caused a serious issue for those of us working with them (the "altitude" of a triangle was accidentally typo'd as "attitude," which caused
significant distress and confusion for several such students; we eventually had to get an official response from the professors specifying that
yes, the book is actually wrong
this one time, don't worry about it too much.)
For folks genuinely completely new to tabletop roleplaying--which is way more likely than it used to be, what with 5e's rapid growth--questions like this are going to be very natural. These folks are
going to be more similar to the adult learners noted above than they will to seven-year-olds first being introduced to the concept of equations.
The rules are quite clear. The players say what their PCs do, the DM says what happens which includes all checks. You don’t need to call it out specifically because there is no specific rule that overrides the general rule.
And I fundamentally disagree. I think it is extremely important to be clear and specific, even when it doesn't seem all that necessary.