People don't really think in terms of 10s either - that's just societal conditioning based on the usage of fingers to count. If you can accurately picture 1/3 of set, but your number system cannot, it's not really that accommodating of a system.
By the way, you can count base12 on your hands using the digits of your fingers. We just teach kids to use fingers because it's the norm.
Why are you so obsessed with thirds? If you care about short, clean fractions on a human scale, base 6 is overall better, see this table: https://youtu.be/qID2B4MK7Y0?t=992
Plus you get all the convenience of having five digits on a hand and 5 as the highest counting digit, so you can count to 55 on two hands, and all the rest of the benefits in that (totally serious) video.
You've got it backwards. Counting to ten on your fingers predates the concept of our modern positional number system by hundreds of thousands of years. That's why base 10 was already the well-established natural choice when that system was invented.
Thinking that you could represent the quantity IIIIIIIIIIIIIIIIIIIII by the string "21" because it's 2 * 10 + 1, or similarly as "three fingers on left hand, three fingers on right hand" seems obvious to us now, but is actually pretty recent technology relative to the whole timescale of human development
I'm not sure what your point is. A numbering system based on five-fingered hands is still base6. It doesn't make mathematical sense to argue that we learn base10 today because our hands have 5 fingers. Even if you argue that we have 10 total fingers, that's base11. Does anybody advocate that? How did we come to such an arbitrary thing as base ten?
No, counting on the hands is not base anything, because positional number systems did not exist. Just like Roman numerals or tally marks aren't base anything.
People counted on fingers like with tally marks - they could express the numbers one through ten. Thus ten became an important and familiar number to humans. Thus it would have seemed natural to use a base 10 number system many millennia later.
By the way, you can count base12 on your hands using the digits of your fingers. We just teach kids to use fingers because it's the norm.