I don't understand why LinkedIn haven't stepped in to this market more strongly.

They can verify if people are real. They have the information on what experience people have. If I was them then I'd be saying that your LinkedIn profile was your CV, and persuading companies to make LinkedIn be their route to applying for jobs.

(Not a shareholder, don't really use LinkedIn myself, just feels like such an obvious step for them to take.)

This is an interesting take. I think they are heading down this path, especially with the 'Easy Apply' button which is essentially what you're talking about.

If I have to guess, the current business model is likely profitable & any deviations too far from that could hurt their bottom line. Incrementing slowly overtime to LinkedIn replacing the CV seems to be the play, mainly to keep the current cashflow going.