That's what you're supposed to say when terminating someone. That meeting is not the time to discuss and debate the merits of the decision, at least from the company standpoint. If the employee is blindsided by the decision, either it's a bad company, a clueless employee, or both.
In your case, it sounds like a bad company but at least you saw it coming.
A good company provides honest, direct and timely performance feedback. The only reason to document it is to provide a defense if the company is later sued for discrimination, which again is practically speaking only relevant for minorities and sometimes women.