I agree that it would make both the drivers and constructor's championships more representative. You wouldn't win either just by having the best machine (WDC) or the dominant pilot (WCC).
Of course as the season evolves, you would get a car that has developed, or take on a driver with an injury, but over all, I think that would come out in the wash.
The driver's market would be weird, currently there's a 'free' market where teams hire drivers they feel deserve to race in F1 (either because of skill or sponsorship pockets). You'd need a new way to decide who the drivers are, and also a way to force drivers out who aren't up to snuff, and promote drivers who have earned a shot. If done fairly, this might eliminate pay drivers from the sport. Of course, without pay drivers, quite a lot of investment in the sport might evaporate.
I love the idea, but honestly, the current system, where drivers are part of the team is how the sport should be. We've seen Hamilton adjust to Ferrari, and vice versa. If they're just hired guns sitting in the cockpit over a weekend, we don't get that evolution.
I suppose most drivers aren't a fan of being corporate shills, and if you're not attached to a team, there would be far fewer sponsor events to endure. Maybe that's what Sainz REALLY wants.