Which arguably suggests we didn't settle on base 10 because we have 10 finger as seems to be often told. We settled on base 10 likely because of politics (in the broader sense of the word)
I imagine there's a natural gravitational pull toward base 10 from having fingers, and throughout prehistory and early civilization occasionally systems deviated to suit certain purposes (like sibling comment about even divisibility of 60) but usually came back to using 10. We've always needed to count, been smart enough to count, and had 10 fingers readily accessible to count, so I wouldn't count that theory out :)
I think the biggest strength of base 10 is not hand-counting (the OP and the Babylonian/Chinese base-12 method are both superior in that regard), but ease of performing pen-and-paper operations. You can literally teach a 6-year-old to multiply huge numbers effortlessly.
Couldnt you do the same using two additional digits? Say binary and hexedicmal arithmetic are as easy as decimal if you substract the bias of being used to decimal.
If you use any other base for writing numbers down, it's just as easy to perform pen-and-paper operations. The only problem with larger bases is that the multiplication tables increase quadratically. Whereas a base-10 multiplication table has 100 entries, a base-16 table already has 256 entries.
Not quite! You can safely ignore identities (0, 1, and 10 itself) so you only have 8 numbers in your table. And multiplication is commutative so you only need 8+7+6... (= (8+1)(8/2) as per Gauss) = 36 entries.
Base 16 would have (14+1)(14/2) = 105 entries. So proportional to base-10, actually slightly harder than you said.
This video convinced me that base 6 would be even better for simple pen-and-paper math, as well as just about everything else: https://youtu.be/qID2B4MK7Y0
It was a long time ago, so not likely anybody wrote it down. But wasn't it from India? Counting-sticks in boxes, when you got to nine (maybe all that would fit in the box?) you put one stick in the next box and 'cleared' the lower-significant box. Apparently zero is a drawing of an empty box...