So from what I understand some particularly fervent Arya/Gendry shippers have interpreted ‘you’d be my lady’ as a sort of quasi hypothetical proposal?
Which it isn’t.
Gendry’s bastard born, an orphan, a former armorer’s apprentice, raised in Flea Bottom and lucky to have escaped by Robert’s generosity. Before the ambush by the Lannisters, Gendry was going to be made to join the Night’s Watch, to sleep and eat and fight beside rapists, thieves, murderers, and other unfortunates. He’s all too aware of all of this. He knows that it’s an impossibility for him to be close to Arya as a friend, let alone have a more intimate relationship when she’s old enough.
Arya doesn’t, however. Since she was Arya Underfoot, she’s run about with servants’ children and her friends have been commoners, workers, themselves part of or children of the laboratores (those who work). She’s accepted this as not exactly normal, but tolerable behavior for a high born lady of House Stark. She wants to be friends with Mycah and keep in his company, so she does. No one has been able to stop her. And I seriously doubt Mycah and her other friends did not express their reservations and discomfort and she assured them it was fine because to her it was. To her they were just friends.
But that’s not the reality of this world. That’s Arya’s tomboyish fantasy, where everyone is equal if she wants them to be and she can enjoy the company of whomever she likes, and spurn that of those she doesn’t like. Arya is being incredibly naive and even a bit cruel (obviously without meaning to be) by offering to be Gendry’s family, in any capacity. Because he’s not a Stark. He’s not even minor nobility (well, he’s Robert’s bastard, but he doesn’t know that). It simply cannot be, and he knows that, and no small part of why he is leaving is because he knows that Arya can’t be family, no matter how hard they try.
She will always be his lady, because that’s how they were born. He will always be subservient in a way that makes him distinctly uncomfortable. Arya thinks it’s stupid, like she does a lot of things, but she’s never taken the social hierarchy as something that just is, whether you want it to be or not. She thought she could overcome social conventions by sheer will and refusal to follow them.
And maybe that worked in the household of Lord Eddard and Lady Catelyn. But she’s not in Winterfell anymore.
In a way, I think Gendry thinks he’s doing her a favor. He doesn’t want to let her get more attached when he knows it can’t last. He knows she’s still a hostage bound to be ransomed by her brother, a king no less, and he’s got very few avenues open to him. The one that offers the most freedom and the least social restrictions is the Brotherhood. So he will join them, and be part of a family at last. A family that is more than just optimistic fantasy.
tldr; he’s definitely not suggesting that he could be Arya’s ‘lord.’ He’s doing the exact opposite, pointing out why it could never be as she wishes, and this separation is the best thing for both of them.