3 car teams are supported by the bigger teams because it allows them to maintain their inequitable position of receiving extra money just for their history and turning up. They are as flush with money as the smaller teams even if they didn't score a single point this year.
Even if a middle team such as Force India has a great year and beats McLaren for example, they don't even get the same money, let alone more, because of the bonus payments to those bigger teams.
These bigger teams pay lip service to the financial problems of the sport but need to give up those payments for a solution to be found. The constructor bonuses can still be paid out according to position thus rewarding performance, but the pot needs to be made bigger.
If it isn't and the big teams just let the smaller ones go under, what we might have is a grid that could be smaller than 18 or 16 cars. Only McLaren, Ferrari and Williams could probably be counted upon as dead certs to turn up. Red Bull would probably stay but may well decide that 6 cars is excessive and run 3, (2 top drivers and a protege) thus taking a car off the grid and a spot for new talent. That's about 12. Mercedes takes it up to 15, assuming they stick around for another couple of years. Who else? Renault maybe, but they come and go (See below).
Then we struggle. Enstone, Force India and Sauber are already having a hard time fielding 2 cars let alone 3. How long would they last with 3 car teams?
What's worse, manufacturers come and go. F1 is inseparable from their marketing arm. Once the costs outweight the gains, they leave, even if they are doing quite well. And in times of dominance their exposure is going to be less if the dominant team locks out the podium. All of a sudden the big teams that thought 3 cars were a good idea don't like it so much when they aren't getting silverware.
In the heyday of the manufacturer era in the 2000s, there were not more than 6 manufacturers that competed as constructors rather than just engine suppliers. If F1 couldn't attract more than 3 manufacturers as engine suppliers under the current regulations, I really don't see how they are going to attract more full constructors than they have now, with the extra investment that goes with it.
Yes, F1 had 3 car teams back in the day. It was also an era where privateers could field their own single entries and the grid was boosted by these privateers, giving lots of up and coming drivers, and yes rich hobbyists too, their start. In other words, 3 car teams back then were not an indication of a formula that was struggling to survive.
F1 could also be more creative with its rules on sponsorship. Why does a team have to paint both its cars the same? Especially now we have the permanent numbers and the silly rule on helmet designs. If a team wants to do a BAR and paint each car different, great. Attract more corporations by giving them more options for marketing. Heck, we might even get some more colour on the grid, what's not to like? Allowing single car entries could help, too. Attract those who want to try but can't afford to run two cars. If they do well with one car, maybe they can attract investors and get enough funds together to run a second car.