It probably helped that their movies were made in an era that didn’t tend to show realistic kissing, let alone more intimate embraces. (It may be that the Code actually prohibited realistic kissing.)
But Astaire’s dislike of Rogers, and mistrust of her ability to carry out his choreography, actually made those movies more romantic, I think: when they’re dancing, he gazes at her so intently that it looks like love. (It’s actually scrutiny intended to catch her screwing up the steps.)