Hi--A couple thoughts:
1. Could it be blossom end rot caused by calcium deficiency and/or uneven watering/rain conditions? I had a lot of that last year myself.
2. You are NOT the only one on earth who can't grow zucchini!! For some reason, that is literally the most difficult thing for me to grow, too!
3. If you think it is pollination, maybe consider getting mason bees. They are relatively cheap ($30?) and are supposed to be 100 times more efficient at pollinating than even honey bees. I was not seeing pollinators this time of year last year either and actually got 2 hives of bees for this reason, but if you don't want to mess around with hives, mason bees are supposed to be easy, cheap and good.
Good luck! May we both have enough zucchini this year to be giving the stuff away!