A system like this is too "forced" for me, personally.
Consider a wizard is casting fireball and a fighter wants to move in to attack. The fighter would have to move on 2, putting themself in the fireball potentially, or wait until 5 and move and attack all at once (but you post seems like this isn't an option really)?
Yes, if the wizard is going to fireball the area, the fighter would want to hang back until after the fireball has taken affect. If the fighter did not move in the movement phase, they would have unused movement left with which to move on the melee phase, so they could then move in and mop up anyone who survived.
What about the rogue who wants to attack, bonus action disengage, and then move away? Movement comes first, along with the bonus action to disengage, so the attack couldn't happen on 5 since they are no longer engaged with the enemy?
No, the bonus action activates when the Rogue uses it. So, what would happen is that the Rogue moves into melee range on the movement phase, attacks on the melee phase, and then uses the bonus action to disengage, and moves away with any movement they have left.
But if the Rogue was not in melee, and using their bonus action to Dash, then that would happen on the movement phase.
While this might produce a general flow for declaring what PCs are doing, I feel there has to be a lot of adlib finnagling going on.
Finally, personally, I've never been a fan of "side initiative". I fail to see how this...
???
If anything, it seems like it would make things more rote and mechanical.
Obviously it works for you, but I would love to see it in action.
Of course, I make no claim that this is a superior system that works for everyone. I have been in groups where, prior to the first character in the initiative order goes, the whole party puts their heads together and plans out their strategy, or one player will tell another player to attack one particular enemy, because he plans on tying down these others with an AOE. This is not the kind of group that really needs this initiative system, because they are already creating dynamism and teamwork.
But that is not my regular group. What happens in my regular group is that everyone thinks of what to do on their turn on that turn. They may have had an idea of what they were going to do before the other players went, but now the situation has changed, so now they have to reconsider. Each player's turn is an island unto itself.
So in the case of the melee fighter and the fireball wizard, the Wizard maybe thinking "Fireball!", but the Fighter goes ahead in the initiative order and jumps into melee. The Wizard player goes, "Oh well, can't do fireball now," and so starts thinking of a different action.
But with this system, in our group, what would happen is something like this:
DM Iosue: Any movement?
Fighter Player: I'm moving here to engage this enemy.
Wizard Player: Hold on! I'm going to let loose a fireball. Hold your movement until after that!
Fighter Player: OK!
This is the dynamism I was talking about that our group doesn't get with the usual initiative. Because it's the "party's turn", and the action of each phase applies to all the PCs, no one's stepping on any toes, telling people what to do on their turn, but there's freedom for give and take, and light strategizing.