Where does that magical monthly money for the car payment come from? If you’ve paid cash for the car, you can no longer put that money in the bank. It doesn’t seem to have much to do with the problem.
The worked examples above started from a position of $0, but this assumes that you already have the cash (maybe an inheritance or some such).
Looking at mere numbers, you do end up paying more on the loan unless you can get a better return on your investment than the loan (and in some cases, you can). Probably you have to take into account a higher price for financing it, but you still might beat the loan if you’re good enough.
Economically speaking, it’s almost always better to have the money with you now than later. Depends on your circumstances.