I would love a little feature to allow a person to customize via config the chance of quality value as defined by the next_probability value.

{
  type = "quality",
  name = "uncommon",
  level = 1,
  color = {43, 165, 61},
  order = "b",
  next = "rare",
  next_probability = 0.1, <--- this one right here
  subgroup = "qualities",
  icon = "__quality__/graphics/icons/quality-uncommon.png",
  beacon_power_usage_multiplier = 5/6,
  mining_drill_resource_drain_multiplier = 5/6,
  science_pack_drain_multiplier = 99/100,
},

You could, while your at it, create config settings for all the other juicy values in these defines (look for them in quality/prototypes/quality.lua)

Could you explain what this parameter is responsible for?

Lets say you put in quality modules into a machine that makes quality chance 15%
That is, essentially, 15% for an Uncommon
The above code defines the next quality and the next_probability at 0.1 or 10%

So 15% of the crafts hit this uncommon, of those 10% go to the next, and 10% of those to the next and so on. By changing this next_probability you can raise or lower the odds of higher quality.

I think it should be a separate mod. In this mod, it would be possible to adjust the qualities by analogy with modules from ModulesRework. Interesting idea, but now I have no so much time to create new mods. Maybe later

P.S. You can make it yourself :)
If you've found what you want to change, that's already half of the development :D