Learn Turkish: Motoru arızalanan araba bir türlü çalışmıyor. - The car whose engine broke down simply won't start.

The phrase bir türlü means 'in any way' or 'no matter what'. When you use it with a standard negative verb like çalışmıyor, it already expresses a strong feeling of frustration and inability ('it simply won't start'). While you can use the inability form (çalışamıyor), the standard negative is extremely common here because bir türlü does the heavy lifting to show that despite your best efforts, the action is not happening.

How exactly does motoru arızalanan araba mean 'the car whose engine broke down'?

This is the standard Turkish way to express 'whose' when the possessed thing is the subject of the action. First, you state the possessed item with a possessive suffix: motoru (its engine). Then, you use the subject relative clause participle -an / -en on the verb: arızalanan (that breaks down). Literally, it reads as 'the car that its engine breaks down'. Because the engine is the subject doing the breaking down, we use -an rather than the -dık object participle.

The translation says 'broke down' in the past tense. Why isn't a past-tense relative clause used here?

The -(y)an participle is generally tenseless and takes its time frame from the context. Since we know the car won't start right now, logic tells us the engine failure already happened. If you really wanted to emphasize the past aspect, you could say motoru arızalanmış olan araba, but simply using arızalanan is much more common and perfectly natural.