Initiative and it's role in combat resolution in RPGs is curious. By example: let's say you are using a fixed initiative either rolled at the beginning of the fight or as a standard fixed score and you find your character has the lowest initiative the truth of it is you aren't going last in round 1 you are going first in round 2. Order really doesn't matter too much after the first round with any sort of fixed initiative.
Do you used phased rounds where everyone moves... the lower initiative folk get to move after the fast folk, this is a huge tactical advantage unless of course you use zone of control rules that trap folks in place when in reach of an opponent. If you are using zones of control, why does the winner of the round get trapped in place before the lower initiative foe can act? So what advantage is a high initiative score?
Real combat is about controlling (by limiting) your opponents options, a touch towards reality is addressing what initiative actually does for your combats. Initiative as advantage makes some sense and works in this context. Combatants with higher initiative should get a wider range of options or possibly more actions then those with lower initiative.
There's a host of ways this advantage can be addressed. In my MOG campaign having a higher initiative gives a combatant a chance to strike before the opponent (initiative is rolled for each side each round so it isn't fixed), what weapons one can use well depends on ones initiative score (as it's really advantage) compared to weapon used (I use a weapon speed score), if one is able to make multiple strikes they need to have a high initiative to pull it off (no picky math beyond comparing to weapon speed and initiative score for me).
Initiative should be more then who goes first, it can also define advantage (number of actions, actions allowed by example) and initiative can mean more then who gets to declare actions 1st or last in round one.