I have a bit different view.
- The car isn't starting again.
This sentence doesn't work well without the adverb again as the present continuous expresses that the action of the engine not starting/igniting is being repeated in a certain period of time. Let's say you failed to ignite the engine and you got it fixed yesterday. But, the car didn't start again this morning and that's when you use the above sentence.
Let's take a look at another example:
How long have you been running? (I have been running) For 10 years.
"I have been running for 10 years" doesn't mean a continuous action as in "I have been reading a book for an hour". You can read a book for an hour without taking any rest, but you can't run for 10 years. The above usage expresses a repetition of running on a regular basis, i.e. 10 miles per week, or 20 miles per week or month.
Now, most important thing is if your car doesn't start for the first time since you bought the car, you should not use the present continuous. You should use the present simple:
My car doesn't start. The engine doesn't start. *My car is not
starting. *The engine is not starting.
The sentence with * is misleading as it might indicate the car/engine didn't start in the past once or several times.
- The car hasn't started again.
Compared with "the car isn't starting again", the sentence contains a different information. No. 1 means you are trying to start a car now, but No. 2 means the car didn't start in the past (you have an experience of it) and the experience was repeated in the past. It could be from a few seconds ago to a few hours ago. Contrast it with:
The car has started again.
This sentence means the engine was ignited at some point in the past, probably not long ago. Again, it doesn't express the present action.
Compare it with:
The car didn't start 10 minutes ago, but it has just started.
There is a difference in using the "present continuous" and "present perfect".
No. 3. *The car hasn't been starting.
The verbs such as start, begin, finish, etc. are an event verb, however, they are different from other verbs in that it doesn't take a long time comparatively to start/begin/finish something. For example:
*I have been starting to have/finishing having lunch.
You could have (eat) lunch for an hour, but you start or finish it in an instance. That's why it is awkward (even nonsensical) to use the present perfect continuous for those verbs. Contrast it with:
I have been preparing for lunch. I have been trying to start the work.
It could take an hour or more to prepare for nice lunch or try to start something. That's why you can use the present perfect continuous for the above examples.
I hope you can see the difference clearly now.