I think they are too lazy to actually look at your profile, they just do a keyword search and then contact you directly form the search page (you can still do this on LinkedIn if memory serves me correctly), apparently, being bothered to click on your profile and read through it would take too much of their precious time
I suspect they ask for a resume so they can send it to someone else. Extracting a resume from linkedin is work (AKA an opportunity for an application or linkedin feature).
I wonder whether folks are more likely to keep their linkedin profile up to date or their resume.
Every LinkedIn profile has a "PDF" button that generates a pretty nice-looking resume representation of the profile.
There's already applications that build structured data (aka HRXML) from scraped/API-grabbed LinkedIn data.
There's a ton of automation in the recruitment industry.
Don't know how it works in the US (or your country of being), but in my country ID laws expect you have to give companies explicit permission to handle your data in the recruitment process.