So RPGs come from a rich history of war gaming with the saving throw and officer rules being the most direct connection.
Over the years, i've toyed around with different ways of dealing with uncertainty, with chance resolution, and with challenges to a character's actual abilities toward what the player is attempting to do; however, I haven't really seen anything or come up with anything worth while.
One might point to Amber's diceless system or to Nordic LARP as a shining beacon of immersion over systemic representation, but those are extremely specific and contextualized instantiations of things not like other RPGs. Most of the attempts feel foreign. It may be that you cannot escape dice if you want a game to feel like a ttrpg.
Do you have a favorite resolution system that doesn't rely on dice, cards, coins, points, or other forms of those things?