I don't see any problem with your house rule, but I'm not sure it's really applicable. Part of the problem is terminology--there's a difference between a round and an action or "turn."
In any round, people take their action according to initiative. If my mage character is fighting an orc, and I win initiative, I get to act first in the round: my action is my turn. Suppose on Round 1 of the combat, I try to hit him with my dagger and miss. He then takes his turn and tries to hit me with his weapon--that's his turn.
For Round 2, I decide combat just isn't my style and cast Command, telling the orc "Freeze!" He hasn't had his turn yet--so if he fails his Wisdom save, when his initiative comes up, that is "on his next turn." He loses his action that round. So the rule has the effect that you want, at least as far as that goes.
I'm not finding anything in RAW one way or another, but it seems to me that any action that the orc could take, including fleeing combat, would be covered by that Command as long as he failed the Wisdom save.
In addition, to my eyes the Opportunity aspect is a gap in RAW.
The Command spell has other examples can effectively put the target "out of combat": for instance, "Drop" will make him drop his weapon, "Grovel" will make him fall on the floor. But as written, the target would be able to take a reaction that round, and then freeze, drop, or grovel the next round.
To my mind, that makes no sense, so I agree with you that the rule change is a good one. If I were GM, I would write that rule change so that if the target fails their Wisdom save, they cannot take a reaction until their next turn after they have performed the Commanded action.
This does not unbalance the game, so long as you remember that enemy casters can also use this spell, and use it to the party's disadvantage.