What would change if there was no rolling for initiative and instead, whoever had the highest bonus to initiative went first (in descending order)? The only rolling would be for determining who would go first if your modifiers were the same.
Citing the Dungeon Master's Guide (p. 270), it is listed as an optional rule:
With this optional rule, creature don't roll initiative at the start of combat. Instead, each creature has an initiative score, which is a passive Dexterity check: 10 + Dexterity modifier.
By cutting down on die rolls, math done on the fly, and the process of asking for and recording totals, you can speed your game up considerably—at the cost of an initiative order that is often predictable.
Essentially, you're creating tiers of initiative. The Alert feat would become exponentially more valuable as an "I always go first" ability, rather than an "I often go first" ability, as would the Barbarian's feature Feral Instinct, which would raise your passive initiative score by +5. Jack of All Trades would make Bards an excellent class for going first in battles, too. Having disadvantage on initiative would almost guarantee you went last.
Since we are using the option of rolling to determine ties, the speed benefit mentioned in the DMG is largely dependent on unique initiative scores. If you have eight combatants in the battle, and their initiatives are +0, +0, +1, +1, +2, +2, +3, +3; you will still end up rolling 8 times to determine turn order. This also means that the higher the amount of participants in the conflict, the more likely you'll have to roll initaitve anyways to break ties.