Well I agree but also think it's a tiny bit more complicated than that. One of the things that makes JA so fascinating was that she was writing at a real time of transition to attitudes to marriage and love. The idea of love and romance and choice of partner on the grounds of 'love' was only just becoming an embedded idea, as opposed to marriage being a purely property and financial based transaction. So in the royal family and high levels of the aristocracy in the late 18th and early 19th centuries marriages were still largely arranged, but in the type of people JA was writing about, the educated middle classes and minor aristocracy, while of course money and class was a very important factor, both men and women were considered to legitimately have an element of choice in who they married (only within quite narrow constraints of course). The idea of choosing a husband for a daughter or ward and instructing her to marry him was considered very old fashioned by this point and everyone is still negotiating what that actually means as of course parents still have opinions (don't we always) and are working out the boundaries of what they can/can't acceptably force through.
So yes, if Col Fitzwilliam had wanted to marry Georgiana, that probably would have been considered within boundaries of him not being thought of negatively or as a fortune hunter (not a great match for Georgiana who would probably have reasonably behaved expected to marry someone with more money or a title or at least an elder son who would have inherited the estate, but acceptable). But I don't think it would have been that he approached Darcy, asked to marry her, Darcey agrees and that was it, his wish would have been Georgiana's command. It would have been more like they got to know one another, liked one another (or least she is persuaded to agree to a marriage), then permission would be sought from Darcy and everyone is happy.
I'm sure you're right that Georgiana with all her meekness and respect for Darcey would be more encouraged by at least tacit approval from her brother before allowing herself to like or fall in love with someone but at her class and status level it wouldn't be an arranged marriage as such. Hell, had she really wanted to marry someone like Wickham, virtually penniless and with only a small salary from a profession to rely on (but without all the lies and profligacy and other misdeeds), she probably could have got permission from Darcey to do so (after all he himself chooses a penniless girl from a vulgar family so hypocritical of him to have refused the same to his sister!), or she could have waited to be 21 and marry without permission (Georgiana as depicted probably would never have done something so bold but other girls did).
I find it really interesting as a study, this era where girls (and young men) had this new freedom and romance in their lives that previous generations didn't, to have at least some choice in spouse, and how all that plays out, some characters still basically just choose based purely on money or status or what their families want, whereas others are fully romantic and choose only based on love or attraction, and how that works out for them...