The stat dominance is definitely one of the things I don't like about the system, yeah.
I have yet to tweak it and run it through play test, but I believe AKAramis already has for a very similar version.
Basically, the tweak is that you leave the task roll largely as is, except skill-levels
subtract a difficulty die instead of giving a +1 Mod to the task number. So skill is pitted directly again't difficulty.
I haven't looked at it in a while, so I forget whether I had determined that 2D6 or 3D6 worked best for the Standard difficulty throw (and whether that was for T4 or T5). So if I use T4 skill levels (which are comparable to CT & MgT) and choose the Standard-difficulty task throw as 3D6 for my illustration (and I always add in the "half-die" or D3, for good measure for all difficulty levels; it can be used or ignored for a number of different things) . . .
If you have Skill-1 in a relevant skill, it is now 2.5 D6. If you have Skill-2, it is now 1.5 D6, etc.
If it is Difficult or formidable (4.5 D6 or 5.5 D6 base roll), higher skill subtracts dice as above. You are still aiming to roll under the target number of [STAT + Mods]. Skill-0 (Familiarisation only) means you are rolling under 3.5 D6 for Standard tasks, so you probably only want to be doing Routine/Easy stuff (2.5 D6 Base roll).
The half-die (0.5 D6, or D3) result I can secretly or openly retain or discard, or otherwise use something from it to make a decision as GM. It also allows me to have a special success/failure roll (either all "1's" or 3 "6's") even on a 2D6 roll, as long as a full proper D6 is being used to calculate the D3 ("1" = 1-2, but you need a rolled "1" to count towards special success, and ditto for special failure with "6's": 3 rolled "6" on any number of dice is special failure; 5-6 = "3" but a rolled "6" will count as one of the "6's" for the determination.