Opinion: will you be reluctant or eager to switch from 5E to 6E?

[OldTrees1]

3d6 works well to that end: extremes exist but are exceedingly rare.

The other option is using some "degrees of success"-system (snip a "damage" step implementation)

Depending on what exactly bothers someone about 5E's ability check math, these can be anywhere from perfect to orthogonal.

For me 1d10+5 works better than 3d6 because I am more worried about the range (compared to improvement) than about the frequency of the extremes. So deleting the extremes might be better than keeping them around at low frequency. On the other hand some players really love their bell curves as ideal solutions.

There are many types of degrees of success systems (some are just "beat DC by N"). Your example sounds like a damage check step. This is great for opposed checks or other cases where you want to minimize improvement on the initial check without minimizing overall improvement. It is another solution I have considered.