Hello, StillKicking, and welcome to GE!
I fully agree with Gustavo's answer and think he has done a nice job of clarifying the singular subject of the relative clause. It might also help you and your colleagues to consider that "systems whose performance" is a different (less wordy) way of saying "systems the performance of which."
Either way, whether you say "Their performance is deemed critical" or "The performance of them is deemed critical", "performance" is the subject of the clause. The fact that the relative clause is modifying a plural noun ("systems") is irrelevant to the choice of a singular or plural noun within the clause.
The subject of the relative clause of your example is not the same as, and is not even co-referent with, the noun that the clause modifies. "Whose" is a possessive subject. Notice that the possessor of a subject containing a possessive never controls subject-verb agreement within the clause. Only the subject head does.
- Their car is nice. [singular subject containing a plural possessor]
- Their cars are nice. [plural subject containing a plural possessor]
- His car is nice. [singular subject containing a singular possessor]
- His cars are nice. [plural subject containing a singular possessor]