Although I do like applying damage to other stats where appropriate, I don't actually think you need to if what you're worried about is balancing them.

STR is the more important attribute if you're consistently getting into combat. All these games share an ethos that combat shouldn't be a hugely frequent thing at the table. In that context, the stats are a lot more "balanced". DEX is by far the most called for Save, in my experience, plus it's how you go first in combat.

WIS/CHA is a little trickier, depending on your individual campaign. Although in Mausritter specifically casting spells can cause WIL damage.

FWIW this is something I grappled with a bit for my own Odd/Cairn hack (slightly exacerbated by some other rules changes), and I eventually came to the conclusion that I didn't need rules changes to fix it. The only thing I really plan to do is make sure the included bestiary includes examples of damage to other attributes.

I see the effect but why is it a flaw? Why would you need to „get around“ that?

Characters roll their stats randomly. Some are lucky, some are not. As a player you adapt your strategy accordingly.

It hasn't been much of an issue the times I've run Mark of the Odd games (ItO, QZ, and EB specifically), but one possible solution is to steal from Traveller and let players pick which stat is damaged. Traveller also applies damage to stats, but players can pick which of the three physical stats (Strength, Dexterity, or Endurance) sustains damage from an attack.

Letting players pick the stat that sustains damage could de-emphasize STR while adding a simple yet interesting decision, but one obvious issue is that it theoretically triples the HP of the player characters (since players won't willingly choose to have a stat hit 0).

I do agree with AwkwardTurtle's post about balancing the stats, and it's probably the best way without rewriting how damage works.