In my Player's Handbook, in the Ability Check section sub header " Using Each Ability ", sub header "Dexterity" it straight up lists Initiative.
Jack of All Trades says " 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.
Initiative checks do not normally include your proficiency bonus, and is an ability check that you make, there for you add half your proficiency bonus.
It's really simple. If you don't like it, as we talked about before you can of course not allow it, but the rules do allow for it. It doesn't really matter what Sage Advice said on the matter ( though they did clarify it ).