The primary difference between the dice mechanics in Numenera and Fate is not the dice you roll, it's when you get to spend resources to aid that roll. In Numenera the system is front-loaded before the roll which led to the frustrations you all felt. You expended resources to improve the chance of success, and when the die roll failed anyway, you rightfully felt that the resources were wasted. In Fate you roll the dice first, then can spend resources to ensure success or at least mitigate failure. This lets you use those resources way more efficiently. Not only do you not waste them on bad rolls, you don't waste them on good rolls either (another facet of the front loading of Numenera's system you didn't bring up).
Argh - the comments are wonky on my iPad...So my suggestions for future games of Numenera would be to simply allow folks to expend effort after they roll the d20.
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And When you spend xp for a reroll you, if the reroll is 10 or less, you add 10 to the result ( effectively you are then rolling a d10+10)