Here's an analogy (maybe not a good one):
Most of us renew our vehicle registration with little complaint because we can just mail in the form with the fee, and you get your new sticker in the mail a few weeks later (at least that's how it works in CT - I recall in NJ having to actually get your car inspected, which always meant long lines at the end of the month).
With driver's licenses, you have to appear at the MVD (or AAA in CT) and get a new photo taken. A little more onerous, but we still do it.
But imagine if you had to go through all the paperwork of registering a newly purchased car every time you had to renew the registration, or take a driving test (or worse, sit through hours of Driver's Ed classes) whenever your license expired. I'd wager a large number of people with expired licenses would be driving unregistered cars.
That's what the directors of most mom & pop races might feel - the odds of someone setting a record in their event is probably far lower than getting pulled over by a cop for driving without a license. It's a risk they're willing to take.