Absolutely, put her on a bike that's good enough that the bike will not be the reason she dislikes riding (if that's the case).
It needs to be ABSOLUTELY comfortable. Fit for her is probably going to be even more important. Fit needs to be perfect.
I understand the need to not spend so much that you're not going to break the bank. But if you want her to actually ride (and not just take it for a spin around the block once a year), it needs to be comfortable enough and it needs to work well enough that she has fun.
I got my wife a ~$500 rockhopper several years ago. She wanted to ride with me at the time, but didn't know if she'd like it. I wanted her to have a good enough bike to enjoy riding, but not be out a ton of money if she didn't like it.
5yrs ago, $500 worked as a price point. Not sure how inflation has changed things today, but it may be different.
I've put some disc brakes on it and upgraded the wheels and drivetrain on it in the ensuing years. The brakes are more confidence-inspiring for her than the stock cheapo vees ever were. They're BB7's...inexpensive, but very functional.
I'd say that weight is a big concern. You don't want the bike to be so heavy that she can't handle it. The earlier guy's suggestion regarding weight is a bit off. His girl sounds like she's now quite a skilled rider, and she's going to have the strength and endurance to match. Given the sort of riding she's doing, a heavier bike is going to be a requirement.
With yours being a raw beginner, it's a different story. I dunno what sort of exercise history she has, but she at least needs to be able to pick the bike up in case she reaches an obstacle she cannot ride or is afraid to ride. Loading the bike onto the rack is important, too. If she ever wants to ride on her own, will she be able to load/unload the bike herself? That's a concern.
My wife is unsure she's ready for a nicer bike yet. She feels that her bike is good enough. However, she does have difficulty with the weight of it. The fork is a large part of that. I do think if I find a better (lighter) used fork, I'll probably get it for her to help with that concern. But when she's ready for a new bike, you'd better believe I'll be helping her choose one.