Here is the thing Zig...
The parent who is not entitled to claim EIC (or other tax attributes) hurries up and files electronically as soon as they can and beats the parent who is really entitled to claim EIC. Therefore the entitled parent's return rejects electronically because the dependents have already been claimed for all tax attributes, including EIC.
If the entitled parent is using a tax professional their options to resolve the situation will be explained to them. If they follow their options they will eventually get the EIC they are entitled to receive, but it could take months to receive it.
If they are online filing, on their own, they likely have no idea that they have any options or how to exercise them if they realize that there are options.
This is something that I deal with on a regular basis. I have even testified in family court many times about this issue.