Not in my system of logic it doesn't. All that page says is that these are the conditions of entry for Executive Club members. I don't see anything that excludes non-members.

Anyway, in practice, I can say that I have never had a problem accessing the Concorde Room simply on the basis that I am flying in paid F (or to be strictly accurate, A) either with my CX Emerald number in the reservation or no number at all.