Well, in earlier editions with 1 minute rounds I would just use a die roll (d6 in 1E, d10 in 2E). The idea being that fast reflexes (high Dex) aren't going to likely matter during an entire minute of combat. I thought of doing it with 5E but with 6-second rounds, things like reflexes, tactical wit, and insight (Dex, Int, Wis) could affect how quickly someone acts.
Between Dex, Int, and Wis, many characters have at least one of those at 16 or higher, getting a +1 mod on the d6. Some don't, so no mod.
I work Alert to usually at another +1 (but many times +2), but we still calculate the normal total modifier, and divide by 3 for the new modifier. So, if a character uses Int 14 (so +2) but also Alert (+5), the total +7 divided by 3 is +2 on the d6. The new modifier is always rounded down. So if a character's normally +8 to Initiative, he is still +2 (8/3 = 2.67, round down to +2).
thats all fantastic and i wish we had time for you to add five more paragraphs about how simple math works but the final nutshells are it would seem one of the three...
1 - It basically is a bunch'a math to reproduce the same results as the regular rules. (Does not seem to be what you are claiming.)
2 - It boils down the difference in character to more character having the same modifiers (less difference shown between characters) and so you give the dice more power and the character traits matter less.
3 - It boils down to create more differentiation between the characters (fewer have the same modifier) and so the character traits matter more.
Now, it seems to me that with all your math involved it comes out to #2 - 3 pt brackets of 5e modifiers (equivalent of six points of ability score) divide down to 1 meaning a lot less distinction between PCs and more of this just being a "die decides".
We tend to go the opposite way - favoring elements that make it more about the character traits than about the dice over ones that buff down the character traits to being more similar between characters.
But thats us.
I have played in games where the modifier was always just a single d6 and where "traits" boiled down to "just big stuff" so a lot of the small differences were not designed or desired in the mix - "don't sweat the small stuff" (DWSS) design - and while it was fun the one area it definitely showed some issues was in the degree of "mechanical" differentiation between the characters being a smidge lower than we liked - but then it was more narratively driven and things like "sequencing events" was not random at all.
My most recent concluded 5e campaign (18 months beginning to end) used "initiative by choice" - designed to remove random order altogether and empower choice not rolls.
Start of each combat the players were told to pick "first or last" and if we did not get a quick answer of "first" the answer would be last.
if they chose first, the one who did so went first. if several chimed in with first, they made a quick decision of which went first or they went to last. After that it was "one of mine, one of yours" repeating until the side with last was down to their "last guy" as that guy was held off for last. So if the numbers were uneven there might be a cluster of multiples from the more numerous at the near-end.
But each time "one of yours" came up you chose which one. So you go first and want your cleric to go later and your fighter to rush ahead while your rogue lets the fighter engage to get sneak - all good. But, same for the other side.
If you chose last, one of my guys goes and the same sequencing by choice went.
After the first turn, the order was set and carried from round to round (due to there being issues with changing sequence order and the turn-based timing all through 5e mechanics.)
It played out great and the players loved it (and at times hated it) for the freedom it gave them to actually *choose* their sequences and order and how it made that opening turn more like the opening of a tactical game with repercussions that carried all through the fight - driven by choices more than dice for order ever did.
So, for
"streamlining play", it removed all the dice rolling for turn order stuff and moved us immediately into "make meaningful choices" (to us that means "the fun stuff") right away.
I represented features that gave advantage on init and such by letting that player choose for his character his own "first" or "last" separate from the group decision. So, in theory, if the group chose first and they did too, the advantaged guy would go, then the party "first" then we went to the back-n-forth.
But thats how we normally tend to look for "what changes do we want to make" -
do they put more spotlight and import on differences in characters, choices made or dice?
We practically never (read this as "never but faulty memory") implement things that spotlight the dice more and the character differences and choices less.
When we want to save time in a mostly dice driven step - its usually by just removing the dice not changing around the dice and mods to create a different dice soup - especially if it makes more "ties" happen.
Why roll dice with a good chance to leave you at "ties"? You were at a tie before the roll, then you rolled and now you are at a new tie? Wasted time.
but again, thats just us.