Like others have said, I think the wisdom of a person (male or female) hiking alone depends on various factors, including of course their back country skills.
And on far to many occasions I have seen confidence that far exceeded the skill to back it up.
Anyone deciding to solo is increasing the risk to a degree. And that's fine. But when it involves another (or others), it's good to take their part of the increased risk into consideration.
As for those that make the declaration that using words like "my" & "let" in this context is offensive & just plain wrong... well, I'm kinda offended at your offendednees
"My" girlfriend/wife is (in it's best & healthy intention) a simple declaration of a partnership. Nothing wrong with that. If it is, then all of you need to quit saying my girl friend & my wife & start saying the wife that I lives with me
If the woman I'm with refers to me as "Her boyfriend", it's a good thing.
I think that "let" is often used as a simple way of saying that that we talked about it & reached a mutual decision about something that we felt was important to both of us.
When a man & woman (or brother & sister or business partners or whatever) talk & make a decision about whether or not to do something, I think that's the way the OP originally intended her question.
In other words "How would you feel about giving your consent for your girlfriend or wife to go it alone out there".
I don't know about some peoples relationship, but when making bigger decisions that will effect both persons, I would hope that she & I would want to reach a decision that has taken both persons feelings & well being into account if we have intertwined our lives.