From the OQ:
To me, "I cannot afford this car" means "I am not able to have enough money to be able to buy this car".
The reason can is used with afford is because people use the word afford as a substitute for the word provision or provide for or allocate/furnish funds for/to.
The common usage, per Wiktionary:
To give, grant, or confer, with a remoter reference to its being the natural result; to provide; to furnish.
Per the original quoted definition, afford is defined as [already] having the money to be able to pay for something. But people don't use that definition when they say "afford". That is to say, English usage of afford in this context is about furnishing of funds rather than having the funds.
"Can you afford to pay for that new widget?" Means: Are you able to furnish the funds to pay for the cost of the widget?
Yes, to afford means in one definition to have the money to be able to pay for something, but it's not just about having the money (to an English speaker). It's also whether the money can be furnished to pay for it.
Why not I do afford this car?
Because do [verb] is almost always used with thinking and feeling verbs, especially for emphasis:
I do like you.
I do like to see this movie.
I do think you're going to like this.
You're not going to do afford this car. You either can afford/furnish funds for the car or you pay for the car.
Q: Do you afford this car?
A: (Do I afford this car [what]?)
Q: Can you afford this car?
I do afford this car is incorrect simply because the statement isn't finished. That is to say, the sentence needs an explicit object that is being afforded to the car.
Rewording for I don't afford this car: no difference in the summary statement. The sentence needs an explicit object that is [not] being afforded to the car.
You don't afford the thing to do or to buy. You afford the means to do or buy.