Here's why your argument fails. This game is simple and there is so many things that supports the No and nothing that supports the Yes.
Everything is broken down in simple terms this edition.
Advantage/Disadvantage
Resistance/Vulnerability
Half/Double
You're trying to get half proficiency on something that doesn't grant full or double and your only basis is the wording of one sentence. Sorry but there isn't enough support to give you a win.
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I have quoted multiple passages of the book, Detailing exactly why JoAT would apply to initiative. Here, I can do it again, in order of pages.
Step 1) Page 7 details the three reasons to roll a d20, Attacks, Saving throws, and Ability Checks. Initiative is not an attack, nor is it a saving throw. It is described in it's passage as a Dexterity
Check.
Step 2) page 54,
Jack of All Trades " Beginning at 2nd level, you can add half your proficiency bonus, rounded down, to any ability check you make that doesn't already include your proficiency bonus". I would like to emphasize the words "ANY" and "Ability check"
step 3) page 72, Fighter's
Remarkable Athlete "Starting at 7th level, you can add half your proficiency bonus (round up) to any Strength, Constitution, or Dexterity check you make that doesn't already use your proficiency bonus." I would like to note that this one specifically says "Dexterity check" in it, just like initiative does.
Step 4) Page 189,
Initiative "Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity Check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of a group acts at the same time." Note that it says "Dexterity Check", not "Initiative check", just like the Fighter ability does.
There is absolutely no reason for Jack of All Trades to not apply, and it even has a similar example in the form of
Remarkable Athlete, which uses the EXACT same wording as initiative does.