Hmmmm I’m on the fence with this one.
I’d usually side with the ‘get your insecurities sorted, the problem lies with you not them’ posters. But I wouldn’t like my DH doing this. The duration, the frequency, the prettiness and the fact she’s opening up new exciting worlds to him. Also, all the smiley positive reinforcement she’d be giving him (rightly). It’s a potent combination. He’d have to be made of steel, and so would your marriage, for the idea never to cross his mind. It’s hard to compete with that, if you’re the person asking him to take out the bins, and the rest of ‘normal life’.
I’d probably find some creative ways to periodically interrupt the cosiness. I don’t mean burst through the door of the lesson with fake emergencies, I mean meet with her to discuss French-related birthday surprises for him, or compliment her on his progress, organise vouchers for the lessons, ask her to translate things related to your holiday house. Things that subtly but firmly reinforce the fact you and he are a team. I wouldn’t see this as acting from insecurity, I’d call it being practical, and protecting my family. No need to do it in a defensive obvious way, just casual and friendly. It’s family money that pays her, isn’t it?
I’m a language tutor myself, and there can be quite a lot of rapport built in this kind of intensive private tuition. I wouldn’t think it strange that a partner might want in, or be curious. Or even a little insecure. I wouldn’t be offended or feel my professionalism was being questioned. I’d be open to reassuring the partner (and hopeful of getting another student!).