Long, long ago, movies from days of the early talkies up through the 30’s featured rich girls running off (Claudette Colbert, heiress in gilded cage fated to marry a rich man but of course she had her true heart set on poor man Clark Gable in It Happened One Night. As if there was any doubt.) And poor shopgirls like Joan Crawford busting out of poverty stricken Coaltown, Pa. for the Big City and looking for a rich man to keep her in style. (whole lot of her movies featured mixing of the classes. Grand Hotel - she was a poor stenographer with a little ‘side business’ keeping company with rich hotel guests.) Gorgeous art deco dream Carole Lombard, her beautiful white mansion, sparring with her impoverished ‘butler’ in My Man Godfrey.
I hate that whole trope. I mean, I don’t mind that some stories of that ilk exist (as referred to in the OP, I mean) but their tendency to be ubiquitous.
There was a book that came out around 1990, Possession by A. S. Byatt. In the book, it’s the guy who is the more uptight and constrained; they have a lot of chemistry and the story is a pretty hot romance, nicely atypical, entertaining. Then they made a movie of it and just had to reconfigure the personalities so that she is the uptight academic and he’s the scruffy worldly male :mad:
Arthur Bach: “I don’t know…money has screwed me up my whole life. I’ve always been rich and I’ve never been happy.”
Linda Marolla: “I’ve always been poor and I’ve usually been happy.”
Martha Bach (Arthur’s grandmother): “I’ve always been rich and I’ve always been happy.”
Okay, I’m glad there are some counter-examples! Mystic Pizza is an interesting case: rich guy, working-class girl, but he’s not into her quite as much as he’s into sticking it to his family. Rarely does that happen: one person is being used and calls the other on it, instead of True Love Triumphs.
Argh! Yes, I only saw the trailer for Save the Last Dance, but they had one shot of the inevitable scene where the black guy is showing the white girl how to keep the beat, patiently snapping his fingers until Whitey McWhiterson masters one-two, one-two. GMAFB.
I actually like that movie. IIRC she was a trained contemporary dancer and she has WAY more skill and interest in dancing than he does. They addressed the race issues on some level in that film, but it wasn’t really a ‘fish out of water’ story for her, either. She made friends pretty quickly and for the most part seemed comfortable in her new neighborhood. There was conflict in that story, but honestly, not that much.
Ha. Of course this makes me suddenly think of Dirty Dancing. I don’t think Baby knew how to dance at all in the beginning, but she was definitely uptight-rich-girl.
I typed a long answer, with cites, even. Chrome ate it. So I typed another answer. Chrome ate it. This is my third attempt. Somehow, I have a feeling it’s going to burn down, fall over and then Chrome will eat it.
Try again! I’m interested.
Presenting The Shirley MacLaine Trilogy
Irma la Douce prostitute with a heart of gold falls in love with an uptight but not rich guy.
Can-Can rough around the edges dance hall proprietor in a love triangle using a straight-laced guy to put pressure on her irresponsible boyfriend. The irresponsible boyfriend eventually gets responsible enough to marry her.
Sweet Charity a dance hall girl loosens up a shy accountant.
Related the idea of the OP are cop procedurals on tv, where iZombie stands alone.
Uptight girl, relaxed guy:
Bones
Castle
Lucifer
The Bridge
etc…
Uptight guy, zany girl:
iZombie
Love Story didn’t follow the uptight girl/rescuing rich guy formula. In fact, Jenny spends most of her time fooling around with Oliver or trying to reunite him with his estranged father. She doesn’t even seem to mind dying very much.
Hmmm. I actually think Dirty Dancing avoided the girl-saved-from-uptightness stereotype. Baby had a moral code (different than being uptight) and lived up to it even when it got her in trouble, and Johnny learned to appreciate that. Dance lessons provided a (genius) backdrop, but the character who arced over the course of the movie was Johnny.
No one puts Baby in a corner!! 
There is a pretty obscure William Holden film called Breezy that is almost creepy because of the age difference, but I think this breaks the mould the OP is talking about.
But curiously forgetting Madonna, who in the '80s played the free-spirit to the rich uptight man in Who’s That Girl, and for variety’s sake the free-spirit to the uptight woman in Desperately Seeking Susan. 
I came in here to mention Pride & Prejudice as an example of a story with an uptight rich guy as the romantic hero.
Lizzie Bennet isn’t poor (although she will be after her father dies unless she marries well), but she’s from a middle class family while Mr. Darcy is a wealthy landowner with family ties to the nobility. While she’s hardly a MPDG, she’s spirited and definitely more outgoing than Darcy.
Sabrina is a counter-example to the OP. The rich leading man (Bogie in the original, Harrison Ford in the re-make) starts out humourless and tightly wound, and gradually becomes more open and care-free by the end, influenced by the chauffeur’s daughter, Sabrina.
In the re-make, it’s typified by two quotes. Linus (Ford) at one point asks Sabrina what people say about him, and she tells him they refer to him as the world’s only living heart donor.
At the end, when Linus is madly trying to catch the Concorde to catch up to Sabrina in Paris, the Concorde agent hands the ticket to the sweaty dishevelled Linus and asks: “First time on the Concorde, Mr Larabee?” He replies: “First time for everything” which nicely captures the disintegration of his neatly ordered life, caused by Sabrina’s love.
In the original, the effect on Linus (Bogie) is captured by a little visual gag in the final scene. Linus always has a neatly furled umbrella. When he joins Sabrina on the liner, he looks around for a place to put his umbrella, and then as a man walks by, wearing coat with a belt, Linus just hooks the umbrella on the man’s belt, then settles down by Sabrina in a deck chair. The man walks on, oblivious to Linus’s umbrella hanging on his coat belt.
Something Wild: crazy Melanie Griffith, and stick up his ass Jeff Daniels. Surprisingly enjoyable.