Here's the problem as I see it. (Please remember that I fiddled around with a similar idea for a number of years.)
In a standard 5e combat, each player gets a turn, and the DM gets one or more turns. This means everyone is sharing the spotlight somewhat evenly throughout the combat. Let's say the group is fighting 4 goblins. As a DM I'll probably have one turn with all four goblins attacking. If there are different kinds of enemies, I might have multiple turns as the DM if I give them each a separate initiative. But the more separate turns I have, the greater share of the spotlight is going towards me.
In a continuous initiative system, now each enemy's turn is going to be split up, interrupted by the players' turns. So I have Goblin 1 moving. Then Fighter does something. Goblin 1 attacks. Goblin 2 moves. Wizard goes. Rogue goes. Goblin 1 moves. Goblin 2 attacks. Fighter goes. Goblin 3 attacks. Wizard goes. Goblin 2 attacks. And so on.
As a DM, I'm taking multiple turns between the players. I'm having to make tactical decisions more often, which is going to slow things down. I just think it's going to result in the spotlight being shared unequally because of the amount of times the DM is going to take a turn.
I don't think Continuous Initiative is a bad idea, but it would require some radical rethinking of how 5e works.