If a recruiter wants to give an employer a list of candidates, and logically wants to give info on those candidates all in the same format, then the CV works. Sure it doesn't work as well as LinkedIn profiles, as CVs come in many shapes these days, all trying to outdo each other in style. But not everyone is on LinkedIn yet, and CVs are still the established norm.

If I had a candidate refuse to give me a CV, and told me to look at his LinkedIn profile, I'd assume he was a dick, and hard to work with. That candidate would be filtered out straight away. Not because I think I'm being too stubborn about wanting to see a CV, hell LinkedIn can make a CV for you, but because the candidate was being stubborn, and rude.

It sounds like you have the advantage of being sought after in this case though, and so some power does reside in your hands to be careless with your approach to job offers. But I wouldn't suggest that everybody takes after your example, for some a job offer is a precious thing, and promoting yourself in the format desired by the employer is a length those people would happily endeavour to fulfil.