Why can't F1 drivers just switch cars from race to race? I.e. 20 races, 20 drivers, 20 cars, everyone races one race in one car.?

Because the Drivers bring money into the team through a great deal of sponsorship. Teams use the Number 1 driver to sell their car through advertising, corporate events, TV and video shows. Its got a downside, some Japanese and Russian drivers got a seat in a car simply because their home recognition pulled in loads of development money, but could not translate to a podium finish.

Its not just about talent, witness the really poor treatment of George Russell once he got a car that showcased his ability, the team wrecked his race .Its Hamilton that pays for that car, not Russell

They could have a claiming race. Such a thing is done in stock cars sometimes. In some types of yacht racing, switching boats is exactly what is done. However, in those contexts the point is to prevent a spendy arms race. In F1 you are supposed to outspend your competition to win. 

Other series have matched cars. Even in NASCAR the big spenders get better results than the "poor" teams. The spendy teams eat gourmet food from their team chef while the poor teams send out for pizza, and so on. So no, it isn't necessarily about the car. 

F1 is as much an engineering challenge as it is a racing series. If you want drivers going in the same car, watch f2 or F3 (both fantastic series). If you want similarly matched cars and boring racing, watch Indy. If you want similar cars and drivers only turning 1 direction, watch NASCAR.

F1 is about having the best package. It's as much about the car designers and teams as it is the drivers, it's why the constructors pays more than the drivers title

Because the drivers aren't independent contractors. They're employees of a specific team.

it's not feasable because each driver is paid  by one team (a lot as it happens)     for instance  Hamilton will be the highest paid sportsman in the UK,   (yes more than all those footballers).. each team works with those drivers   having their own  engineer and mechanics, which grows a bond, a team bond, but don't get me wrong i fully understand the point you are trying to make, well on that note why have Mercedes been allowed to use DAS  (dual axis steering) system?   this is giving their car big advantages, which some of the other teams just can't hope to compete considering the design & development needed and the costs, and was never going to be a conducive decision to making the sport more competitive, rather the opposite...

So get rid of DAS,  get rid of DRS, (Drag Reduction System)  which is in my opinion taking away a big  part of the skill factor,  and ERS  (Energy Recovery System) to be honest....then minimize TCS and if it does exist ban ABS...fed up with them making these cars easy to drive compared to years ago..

and finally for me get rid of these tyre restrictions, which were meant to level the playfield and have acheived anything but..start allowing far more freedon in tyre choice, and start giving the technical cheifs a few more headaches and problems to solve,  f1 has lost so much of it's unpredictibility factor, which i'm sure is what this question is really about..your idea just wouldn't be feasable to some of the teams who would laugh and think it was a joke it they decided to do it, but i think it actually sounds quite interesting, if only it were practical

Because contrary to the belief of many, racing is actually a team sport. It takes managers, drivers, mechanics, and engineers to win a race. Your suggestion would be akin to a quarterback in American football playing for a different team each week.