No, it does not.
The damage dice for the sickle according to the PHB (149) is 1d4 slashing damage and only property is being light. To put things into perspective is best to use the closest simple weapon that fulfill the finesse property, that is the dagger. The dagger do 1d4 piercing damage and has the finesse, light, and throwing properties. Given that both weapons have identical damage, it is safe to assume that the consequences of giving the finesse property would make no difference. But, to try to expand the answer.
Bypassing damage resistance
As far as balance is concerned this is one of the biggest points to address. Given that creatures can have resistance to one type of damage, the question would be, does it make any impact? Well, let see, first we have to analyse which classes would benefit for this. First, we take from the equation any class with the martial weapon proficiency, since there are better weapons for those classes, the monk class because can turn any monk weapon into a DEX based weapon, and any caster class that won't be hitting things on melee. That leave us with the Rogue, Druid, Bard, and Cleric.
Lets reduce a bit more, the druid can use Scimitars, a 1d6 slashing martial weapon with the light and finesse properties. A better alternative than the sickle. So, this leave us with the rogue, bard and cleric. The highest damaging weapon that the rogue and bard can use is the rapier, a 1d8 piercing weapon with the finesse property. And this is where we are going to take into consideration the damage resistance.
Let assume that we have an enemy with piercing resistance (that is, it reduces by half any piercing damage it takes).
Rapier: 1d8 + 5. The average damage of 1d8 is 4.5 so 4.5 + 5 = 9.5 points of damage. Applying the resistance would yield 4.75 points of damage on average.
Sickle: 1d4 + 5. The average damage of a 1d4 is 2.5 +5 = 7.5.
The difference is 2.75 points of damage per hit favoring the sickle. Nothing to write home about, though. That is, taking in consideration just the weapon attack. The real difference comes when you consider the rogue's Sneak Attack, where the damage reduction really hurt the overall damage outcome. The other classes do not benefit from this as much, since they or have some martial archetypes or their main form of attack is not melee.
Therefore, the only class that really benefits from this change is the rogue, but it has alternatives to solve this problems: Multi-classing and feats. Funny enough, the rogue is the class that has more ASI/feats than the norm, second to the fighter, and that really benefits from a bit of multi-classing. So, there is not really a big advantage to the rogue, specially if the player wants to maximize the damage it can do.
Retracting a decision
As the DM you have all the rights to retract a decision on your games. If a change suddenly seems more powerful than expected and its ruining everyone game, you can always take it back.
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.
Best Answer
So first, at least one suit of armor available in the official materials has a helmet included: The plate armor.
(Thanks to nitsua60 for this.)
And that armor doesn't prevent you from wearing special helmets at all. There's nothing in its own description, and nothing in the helmets themselves, that says "cannot use with any helmet item" or "cannot use with plate armor."
But more to the point, 5e, unlike 3.5e, has a very bounded set of attack bonuses and AC. +1 AC is actually rather useful in 5e. So, by doing this you discourage your players from using utility helms in favor of combat power.
There's nothing stopping you from handing out +1 AC helms, but you could also hand out Rings or Cloaks of Protection if you want to swing the difficulty some without limiting utility options as much. And as noted the fluff doesn't match up with the Plate armor fluff.