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Your best bet is a system that uses Dicepools

In Storytelling system, which used in Vampire: The Masquerade, Werewolf: The Apocalypse and other White Wolf settings each time you need to roll, the Storyteller (=gamemaster) announces the Difficulty of the action and skills involved. For example, trying to shoot someone with a pistol is Firearms+Dexterity, Difficulty is 6. It means that the player sums up Firearms of his character (let's say, 3) and Dexterity (let's say, 4): it is his Dicepool for the given check, 7 in this case. If he would try to heal someone, might be Intelligence+Medicine.

He then rolls this amount of D10, and each result that equals Difficulty or is higher is a Success result, each result that is less than Difficulty is a Fail, and each "1" on a die is a Botch. If Difficulty is 6, it doesn't matter if you roll 6, 7 or 10, anything is a Success, and both 3 and 5 are equal Fails.

If there are any successes, you count them, and deduct the amount of botches. That is your amount of success for this roll. If at least one is left, you succeed in the action. Based on how many successes did you get, you may succeed better or worse.

If there are no success dice at all and no "1" dice", the roll result if a Fail. Nothing happens. You missed and didn't hit the target, you failed to repair the car, etc.

But if there are no success dice and at least "1" at the same moment, it is a Botch. Not only did your pistol misfire, but it also jam. Not only did you fail to repair the car, seems like you also broke something beyond the ability to repair it. Not only did you not heal the patient, you also made the wound worse, etc.

Here is my AnyDice program that you may use to view probability curves. It is well-commented, so shouldn't be hard to understand it. As you may see, the bigger dicepool you have (the better your character is supposed to be at doing something), the more is the average amount of succeess dice (better performance) and stability (becomes unlikely to have a Botch).

Of course, I am not forcing you to use Vampire, but a dicepool-based system seems to be what you need.

P.S. You basically mentioned a dicepool-based system in your question:

Count 4+ in (Skill)d6 increases accuracy with skill but precision decreases as skill increases.

But that's actually not completely right. The more dice you have, the lower is your probability to fail the task. Yes, standard deviation increases, but far slower than mean does. Watch the Summary tab. If you change 4 for some other number, let's say, for 2, you may notice that deviation also decreases. Character performance in an easy task is more predictable than in a hard one.

Botch mechanics from Storytelling system also improve the situation. But that's another story.