The thing is, it really depends on the context. First, how did you find the job - through LinkedIn's official application system, or did you spot it on a recruiter's profile? Because if it's an actual job posting with an "Apply" button, I'd say use that system as your main route. It's tracked, it's documented, and LinkedIn's algorithm sends your application straight into the right pipeline. Sending a direct message to the recruiter at the same time, why not, but like a few days after you've applied, not before - otherwise you risk coming across as someone trying to bypass the normal process, and that can look weird.
As for the cover letter, being real: a lot of recruiters at big companies skim the CV first, then the cover letter. But - and this matters - if you send one, make it short and punchy, not 500 words of generic intro. One or two sentences showing you actually know the company and why this role fits you, that changes everything. If you're using LinkedIn's application system, there's usually a "cover message" section - do it well, that's your chance to stand out from the other candidates who just leave it blank.
Have you identified who the recruiter or your future manager is on LinkedIn, just to see if there are any commonalities we could leverage?