How does success chance change when a certain die result counts as multiple partial successes?

Dice mechanism: count wins (> 4) of [Mod]d6 (AnyDice).

• Mod ranges from 1 to 10.
• Difficulty ranges from 1-10, 1 being trivial, 3 being moderate, 5 being difficult, and 6-10 being ever increasingly unlikely at all levels of mod.
• Game provides mechanisms for characters to "push themselves" to gain extra dice when their Mod < Difficulty (otherwise they would auto-fail), so a roll can always be attempted at the minimum possible dice pool size.

When counting wins, any die with a result of 6 counts as two wins. How does this alter the probability of success?

Here is a version of that Anydice program that counts 6 as two successes:

You can now get up to 20 successes in theory when you have 10d6. That means that the probablity shifts to larger numbers of successes. It looks like this:

compared to when you do not double on a 6:

For example, for 4d6, you have lower chances for 1 or 2 successes, but higher chances for 3 or 4 successes, and gain chances to even get to 5 to 8 successes.

For your difficulty classes this means the following success rates (rounded):

Class 2d6 4d6 6d6 8d6 10d6
Trivial (1) 56% 80% 91% 96% 98%
Moderate (3) 8% 33% 57% 74% 85%
Difficult (5) 0% 6% 20% 38% 56%
Unlikely (e.g 7) 0% 0%* 4% 13% 26%

Even with the full dice pool and doubling, getting anything that needs more than 5 or 6 successes is really hard. Getting 10 successes at best has a chance of about 2% with all 10 d6.