The following mechanics would be affected:
- Its possible that a surprised person with a high initiative modifier and/or who got lucky might act first; totally nerfing the use of stealth by their opponent.
- Spending inspiration will negate the effects of surprise totally for PCs
- The ability to take reactions before your turn when surprised is a huge benefit. Being able to take an opportunity attack cuts down movement options for the surprisers and being able to cast a Shield spell is golden.
- You eliminate the possibility that the non-surprised people may act twice before the surprised person acts - this will make surprise a much less dangerous situation all around.
- It would make the assassin roguish archetype pretty useless
Existing mechanics made redundant or obsolete?
Surprise (and by extension Stealth) is effectively redundant or obsolete. It would have almost the same effect and be much simpler if you just eliminated surprise altogether.
What other things grant disadvantage to initiative?
Anything that gives disadvantage to a Dexterity check gives disadvantage to Initiative because initiative is a Dexterity check. Worth noting, in passing, that certain classes "Jack of All Trades" ability gives half proficiency on initiative checks as it is an ability check in which you are not proficient.