They aren't required, it's certainly possible to have a number system and not a writing system. But for anything where you need to do non-trivial arithmetic or accounting you can't just do all that in your head and memorize the results. You need some way of recording the results in a persistent way, even if it's just tallies carved into a piece of wood. And I imagine any culture that figures out a recording system for numbers would pretty quickly (in historical terms, so like years or decades) develop a writing system. I could be wrong of course, I haven't really formally studied ancient history.

Aside: Is Quipu not a writing system? I thought there was some evidence indicating it was.

As far as I know the invention and use of counting boards went hand-in-hand with written (cuneiform) records. Or at least, we don’t have any evidence of earlier positional counting boards, only artifacts like tally sticks and piles of (non-positional) tokens.

The Sumerian society that developed cuneiform numeration/writing was a large bureaucratic urban society built on complicated debt/taxation/social organization.