If I understand your system correctly, it will give rogues an automatic way to activate Sneak Attack:
The green character is the Rogue and the blue one is an enemy. On his turn the rogue steps two squares to get behind the enemy and turns for free. He attacks (with advantage, -2 to enemy AC, and possible more reduction due to the shield), activating Sneak Attack, and then gets to step back two more squares using his leftover movement.
On top of this you should consider things like whether a fighter can riposte to an attack done from the red zone.
Is an enemy/ally in red zone considered to be visible? For example Rogue's Uncanny Dodge says (emphasis added)
when an attacker that you can see hits you with an attack, you can use your reaction to halve the attack’s damage against you.
Can a rogue still activate it while being attacked from behind?
Why would you Shove without a follow-up?
Knocking someone down is not really a good strategy in any fight (real or imagined) if you don't have a follow-up. When was the last time you saw a UFC/MMA fighter push someone to the ground and then just let them get up? When somebody in a gang does a shove, they're doing so to let their friends jump on the fallen.
Once you knock someone down, your next step is really to Grapple them or attack them while they are on the ground. A Grappled creature's movement is reduced to zero, so you can effectively prevent them from getting up (like the gang above)
In fact, the Monk actually do this Shove and Attack starting at level one with their Martial Arts skill. So you knock somebody down (Attack Action), kick them in the head (Bonus Action) and even step away if you want (Move Action, their AoO at Disadvantage). If you move away, they typically can't attack you on the following round because they have to waste their movement standing up.
Note that at 5th level a Monk (or other PC with Extra Attack) could do both a Shove and Grapple as part of the Attack Action to effectively pin their opponent. Escaping the Grapple requires an action and standing requires half movement, so pulling this off pretty much wrecks the target's turn.
This is clearly a gang-up tactic, but that's expected.
Is Initiative the Problem?
Consider that the problem here may actually be the initiative order itself.
You point out the problem of effectively "metagaming" the timing of the action, but also the problem that it will basically never allows you to benefit.
DMG (P.270-1) presents an alternate initiative system titled Speed Factor.
Under this variant, the participants in a battle roll initiative each round. Before rolling each character must choose an action...
This is followed by some modifiers for speed and action type. The action order in each round is random (weighted). This means that knocking someone prone will always have at least a chance of being useful. And there's no "house-ruling" required as this is straight from the book.
If you're really worried about the metagaming aspect of Shoving/Grappling etc. this might help as well. I've used this "initiative per round" and it does kind of solves the problem of gaming the initiative order, both for things like Shoving and for things like Healing. However, it introduces other problems like wasted Actions. So it may be a little much just to solve the Shoving problem.
Best Answer
I wouldn't do it, because Cost Matters
The key component that this idea is missing is the cost vs value of the Disengage action. It is an action, a special ability provided by a class feature or a special ability provided by a magical effect.
But the universal cost of the Disengage is to gain it by spending an action. Everything is (mostly) balanced around this cost. This means that Disengage is intended to be, at least somewhat, as powerful as other uses of your action, such as:
With this change, you are making Disengage much much worse. And you can do that. But you shouldn't until you address the existing cost of it. If you don't want it to be as valuable as the Attack action, change the necessary cost of the Disengage so it's not as expensive as an Attack.
My suggestion? Half your movement to Disengage, or do the ol' 5 foot step rules from prior editions.
Your change also hinders casters and rogues quite a bit, as they have few defensive tools if they are surrounded, and one of their tools is the Disengage action via special abilities. Your melee fighters will hardly use it at all, so expect there to be a power difference between melee/ranged characters, favoring melee, in certain fights.
Overall, though, my suggestion is to not bother with it. There is a strong lack of mobility with players in DnD 5e, due to not needing to flank, few AoE abilities that center on the caster, and weak opportunity attack uses, so melee combat is generally a stagnant mosh pit of damage. If you impact Disengage in this way, combat will move more in the direction that prevents mobility.
On the flip side, melee lines would be harder to break, and units in protected positions would have an advantage.