Part of the problem, sure. "Crux" I doubt.
What percentage of folks trying to call you, do you imagine, forget to dial the 1 ? I seriously doubt that it is 100%. 1+10d dialing has been around a long time. I'd be surprised if the human error rate is much above 10%.
The call will probably complete without the 1 from any major cell phone provider.
Moreover, screening is different for pay phones. Calls from other types of lines might get the right response when they dial wrong. Your testing wasn't thorough enough and "I think" it led you to an invalid conclusion.
I know that for unusual things like this it is likely to be a MAJOR hassle getting the drone who answers to understand the problem but you really should try reporting it to the predominate LEC for that area. Of course, it will be a waste of your time unless you can give them the phone number of the pay station where you experienced the problem ... because they don't know how to take a report without a phone number. And it might be important because it might not be an LEC phone, in which case it might not be their (AT&T/SBC) problem at all.