Any time I see such a post, I wonder why people hated weapon speed so much. Then I remember why and give a sight of relief that it is over and done.
Range weapon can be very slow. Aiming a bow when not in hand to hand combat isn't that stressful. But in hand to hand, your chances to actually aim and hit make you slower than a sword. So in melee range should be a d12...
But for casting... Shouldn't a cantrip be way faster than a two handed sword? If we see cantrip to something akin to the Harry Potter fights, a cantrip should be very fast, earning a d4. And shouldn't we modify the die for tier spells? 1-3 d6, 4-5 = d8, 6-7 = d10 and 8-9 = d12?
And for melee? Shouldn't small weapons allow 1d4? Medium weapons d6 and Two handed weapons d8?
5ed went for this way to do initiative for a reason, it is to make all classes on equal footing and to simplify it. If you really want to add more immersive stuff, going back to the weapon speed factor and casting time might help you better. 2nd edition had a simplified version. It was working ok. Maybe going back to this method would be a better middle ground? Already we see people adding more and more complexity. At some point, I do not think that going back to weapon speed factor should be a thing. But 2nd edition style initiative should work out just fine.