A great rating may not bring you a promotion or a big number bonus, but a "not meet" is not acceptable.
Bureaucracy may play its role when a company has to take actions to shrink its workforce. Occasionnally, restructuring will be executed by a third party consulting service because no one within the company is willing to be in that position. The consulting service is not familiar with the situations and the easiest way they would take is to let staff members with the lowest performance rating go. Downside risks are obvious if one of your performance reviews is "not meet".
Not responding to emails promptly is not uncommon. Some people tolerate such behavior while the others hate it. But anyway, I don't think it is a big deal unless all projects are very time sensitive or a material loss was effectively caused by your slow response. If nothing can be substantiated, a project manager's opinion shall not impact a performance review to the extent of meeting or not.
If I were the "Fish of the Atlantic Ocean", I would escalate the issue to the senior management and let them know my thinking. If they don't want to fire you, they would change the review rating.