I would argue in the OPs case that if the DL is not job relatedand consistent with business necessity that the termination violated the EEOC laws on not holding financial issues against an applicant or employee since the child support arrears are not material to the job either.
You need to understand two things about the EEOC’s positions on things like this. First, the EEOC’s positions are not law and don’t become law until the appellate courts issue opinions consistent with the EEOC's position. The EEOC has a mixed record of success in this regard. Second, it is not federal law (regardless what the EEOC might want) that any reason for termination that is not consistent with business necessity amounts to illegal discrimination and thus is illegal. If that were the law it would turn American employment law on its head an convert our at will employment system to one more like some European nations that require good cause (or more) for termination. That is clearly not the state of American law.
Second, you skipped over a key part of the requirements to make the case for illegal discrimnation under federal law in this sort of circumstance. One of the critical steps in the sort of analysis that EEOC is using in the example you cited is that the policy of the employer must have the effect of discriminating against a protected class even though on its face it appears to be nondiscriminatory. That is, what is necessary to prove first is that the policy of the company would have the effect of discriminating against a protected class. So if the policy is that the company will not hire or will terminate anyone without a driver's license, the plaintiff would have to show that applying this policy would adversely impact some protected class significantly more than another. In other words, one would need to show, for example, that such a policy would tend to exclude, say, one race signficantly more than others to claim the policy resulted in illegal race discrimination, Further, the OP would need to be part of the indentified disadvantaged class to succeed.
Only once the plaintiff proves the discriminatory effect of the policy would the employer need to resort to a defense that the termination of the particular employee was based on business necessity or otherwise justified.
So far as I know, there is no good case to make that one protected class would be significantly disadvantaged by a policy that the employee must have a driver’s license other than another, other than perhaps, a disabled employee. So if the OP were a disabled employee who could not drive because of his disability then perhaps he’d have a claim to pursue. But the facts here do not suggest that to be the case, and I'm not seeing any other possible discrimination claim that could apply here.