ARRL now conducts background checks on ARES volunteers, so the issue is rather moot (in fact ARC and ARRL have an MOU stating that ARRL is responsible for background checks on their volunteers assisting ARC, which presumably applies to this call as well). There have just been too many incidents of people with e.g. sexual assault on their rap sheets turning up working in shelters where there are all manner of vulnerable people. Terrible optics if nothing else.

I suspect the general class requirement is just to cut down the applicants to a pool of more dedicated people. A very real problem with disaster volunteers, if I might be a little harsh especially from the amateur radio side, is "fair-weather responders" who will go through all the training and then not actually deploy. I've also personally found that people with higher class licenses are more likely to be into digital modes which is what they're mostly looking for, but that's getting less true as the equipment gets cheaper (to be honest I've never even seen a packet TNC in person).

I happen to be both an ARC disaster worker and amateur radio operator, but I speak for myself.