It is faster, but it removes the usefulness of a lot of rules. Like weapon speeds and casting times.
If you had an initiative modifier based on your weapon (great sword -2 to initiative) and spell casting costs segments then it wouldn't matter about having more powerful spells at the same level.
It would also give reasons to use smaller damage weapons. I can see a Rogue getting a bonus or advantage to attack or sneak attack creatures that haven't gone yet in initiative order, and a light low damage weapon would allow them to go first more often.
There is a lot of tactical depth that could be added back into the game with this.
Well done!
This right here gives you a reason, outside of fluff, to take a different type of weapon. This also eliminates the use of adding in those stupid rules that this ability only works with this type weapon.