This would never happen, and wouldn't work for a few reasons, but imagine this for a moment.
1. The racers are employed by the league, not teams.
2. The pit crews are employed by the teams.
3. Teams employ their own test drivers.
4. At the beginning of each race weekend (including FPs), drivers are assigned to random cars through a random draw. You can even make a press event of it.
5. FP, Quali, and Race day happen as though the drivers are part of the team for whose car they are driving.
Thusly, the drivers championship becomes a true drivers championship, completely decoupled from the constructors championship.
The viewers get a much better view of who the best drivers are, and also which cars are best.
Teams have no agenda other than to win the constructors championship, and drivers have no agenda other than to win the WDC.
This eliminates jockeying for a better team, which is really a disruptive pursuit, ie Alonso for the past few years...
Once again, this is completely left-field, but I felt like throwing this out for conversation's sake.