"The car is cool."
Both are correct but there's a slight difference. When using 는, it's typically in comparison to something else. For example, with the sentence "The cat was behind the house", you would use 는 if you were comparing the fact that the dog was not behind the house. However, if you were just saying that the cat was behind the house with no comparison, it would be more correct to use 가. However, you could use either one and isn't that big of a deal if you accidentally use one instead of the other.
It's probably just that the (지잖아) form was not added to the list of accepted sentences. When you get something marked wrong that you think is correct, there's a "report" option, where you can choose that yours should've been accepted.
Although, 차는 gives the feeling of a general statement about all cars rather than the one, particular car.
One of the mods explained that in usually, the sound carries over when they are part of the same root word.
So, like the past tense of write (썼어요) will have the sound in the batchim carry over because it's part of the same root, but 멋 and 없 don't share the same root, so you don't hear the sound of the batchim of 멋 in 멋없다.
I think people make an exception with 있다, likely with it being positive vs 없 being negative. But that's a guess