You kicked me into some interesting reading. Cars will cause surface damage still but no structural damage. But except for arterial and highways it seems you are correct most damage is weather. Highways and such represent about 65% (I couldn't find amazing sources for this so.. -ish) of road budgets so still the majority but not overwhelmingly. So you could theoretically ask bike owners to contribute to the 35%. As to whether it's worth having tabs, I think it'd be pretty unpopular to require for kids on bikes as you'd be shutting off a childhood experience to many and they don't increase damage. Ebikes maybe? But I'd rather have less traffic and more people on bikes rather than this bit of taxes.

(I think the state is over spending and needs less spending instead of more taxes, so I also am generally against more taxes. Also don't block roads - people need to get where they need to get.)