Yes, I think that the RF, and no doubt the men in gray, wanted Elizabeth to marry a British aristocrat. Supposedly her parents gave several dances to which they invited suitable young men for her to meet. But she had already given her heart to Philip and was having none of the young aristos.
I posted this on one of the other threads, but my favorite story about Philip and the men in gray is that evidently one of them was rather officiously showing Philip around Windsor Castle shortly after the engagement. Not known for his patience, Philip finally managed to shorten the narrative by telling the man "Yes, I know. My mother was born here."