Here's the base 12 multiplication tables: https://math.tools/table/multiplication/base/12

It doesn't look more or less nice to me than the base 10 version. Can you describe what is supposed to make it look nicer, especially to aliens?

In base 10, the times tables for 2 and 5 are easy because they divide 10. If I want 2*7, I know 7/5 is 1 remainder 2 so it's 10+2*2=14. As for 5x7, I know 7/2 is 3 remainder 1 so it's 30+1*5=35.

In base 12, there are similarly easy rules for 2, 3, 4, and 6. Doesn't seem like that great of a trade off but it could be beneficial. That also just comes down to 12 being a "superior highly composite number".

If I personally was allowed to rewrite our number system, I think I'd choose a base that is either a superior highly composite number or a power of 2. So something in the set [2, 4, 6, 8, 12, 16, 32, 60, 64...]. I doubt 10 would even cross my mind as an option if I didn't have 10 fingers.

Another layer to this is that numbers one-off the factors have nice patterns too. For base 10 those are (2, to include the factors), 3, 4, (5), 6, and 9 sort-of, since it's one below 10. Just think how awkward 7 and 8 times tables were compared to the rest. With base 12 I found that all numbers, even 7 and 11, end up having usable patterns and are easier to count by. Of course, I'm still not used to having 12 digits, but on paper I could tell they would be pleasant to count by if I had learned base 12.

And an even deeper insight is that it doesn't really matter that much and isn't worth shaking up the whole world to change. We're not going to be better or worse at math because of our number base.

You have to fill out the table yourself to appreciate the patterns.