Is a school supposed to offer the financial aid to cover a gap between COA and EFC?

“If no, what should the parents or student do?”

If a school is not affordable, then you pick a different school. Some kids do two years at a community college near their home (so that they can save money by living with one or both parents), and then transfer to an in-state public university for two years to get their bachelor’s degree.

There are at least two types of “not affordable”. One type is where a school does not meet your “need” as calculated by the FAFSA. Another type is where the family’s ability to pay as computed by the FAFSA is not consistent with the family’s ability to pay as felt by the family. Both are common.

Fortunately there are a lot of very good universities and colleges.

Occasionally some time around about March or April a student will find that they do not have any affordable acceptances at all. In these cases they can either take a gap year and apply to more affordable schools the following year, or start at community college with the intention of transferring either after one year or after two years.