Some F1 teams care a lot about the Constructors' Cup, but they don't care about the Drivers' Championship. The best example is Williams, when Frank was alive.
Some teams care about the Drivers' Championship, but not the Constructor's Cup. The best example is Brabham, when Bernie Ecclestone was team owner, and Nelson Piquet was the name driver.
Some drivers don't want a competitive team mate. The McLaren pairings of Alain Prost with Ayrton Senna and Lewis Hamilton with Fernando Alonso, and Hamilton with Nico Rosberg at Mercedes, created tension in those teams.
It's very possible that Max Verstappen has requested a weak team mate at Red Bull, who probably also doesn't have an equally prepared car. Michael Schumacher had his best results when he could count on Ruebens Barichello to be his obedient bitch.