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We've evolved with 14 knuckles on each hand... and a brain that struggles to keep 7 elements at once in operating memory. You can also count up to 1023 with just 10 fingers (in binary). It's not a lack of fingers problem.
I'm not sure what problem you're referring to. I mean if we naturally leant towards base 12, I wonder what would be different, if anything?
The problem is our brains have a limited operating memory. People can (unless disabled) easily track 1 o 2 items at once, even 3, 4, 5... and start losing track somewhere around 6 or 7; 8 is considered exceptional.
That's why kids don't generally use their fingers to count 2+2, but start using them for "harder" operations like 4+4.
Base 10 is already past our brain's limits... but we're kind of fine with it because we can use our fingers (think of it as evolving at a time before formal education when most people were illiterate).
Base 60 is also past our brain's limits, but it's easily divisible into easy to track 1, 2, 3, 4, 5, or 6 pieces (aka $lcm(1..6)$), which makes it highly useful. The Babylonians still used to write it down as base 6×10, and it was common to count on knuckles and fingers as 12×5.
The uneducated populace picked up the easiest part of the two: 5+5.
If we had 12 fingers, we could've as easily ended up using base 12, only thing different would be 1/3 would equal exactly 0.4, while 1/5 would equal 0.24972497... oh well, we'd manage.
If our brains could track 12 items at once however, then we could benefit from base $lcm(1..12)$ or 27720. That... is hard to imagine, because we can't track 11 items at once; otherwise 27720 would jump out as "obviously" divisible by 11, 9, or 7.