Supercapacitors as car batteries
It is not practical.  I do not know why people do this, there is no benefit whatsoever.  It amounts to misuse of something useful.  Simply put, those videos are by people who don't know what they are doing and are misusing supercapacitors for a bizarre and senseless application they are neither well-suited to nor even practical.  And they are offered on the automotive market, just not as battery replacements, for the same reason headlights are offered on the automotive market, just not as car stereo replacements.  Because that wouldn't make any sense.  

The sole reason supercapacitors exist is power density.  They have terrible energy density, and that terrible energy density comes at many many times the cost.  The entire point of a battery is bulk energy storage.  Using supercapacitors to do the thing they are the worst at instead of something that is cheap, readily available, and proven for over 100 years is... the kindest but much too weak word I can use to describe that is "silly."  

Those videos exist, but just because there is a video of it doesn't make it a good idea.  It isn't.  

What is a good idea is using supercapacitors for the reason they exist, which unsurprisingly is the exact way they are being used in automotive applications.  Batteries have great energy density, but compared to supercapacitors (or any capacitor), batteries don't even come close in power density. Beyond that, forcing a battery to provide high amounts of power is hard on it and will reduce its long term life, and the quicker you drain a battery, the lower its apparent energy capacity will be.  A battery will last much longer if drained at a 10 hour rate vs. a 1 hour rate.  Meaning, at a rate that will discharge it in 10 hours vs. just 1 hour.  Higher power means a higher discharge rate.  

This power density weakness is bidirectional: batteries are bad at delivering huge spikes of energy, and bad at accepting them.  They like things nice and steady.  That is where super capacitors come in.  They have terrible energy density, but great power density.  99% of the time, the big power spikes demanded in automotive applications are also brief - things like braking, a burst of acceleration, the inrush current of the starter motor, that sort of thing.  The only reasonable (and intended) way to use a supercapacitor is in addition to a battery, never in replacement of a battery. They perfectly complement each other.  A battery deals with storing tons of energy, while capacitors deliver it at high power when needed.  They permit things like capturing nearly all of the energy back from regenerative breaking, because all that energy can simply be dumped right into them and they'll handle it like champs.  It can then be siphoned back into the battery at a controlled rate that the battery can deal with. 

Supercapacitors can let even an extremely weak battery in extreme cold start the car, because the battery is relieved of power demands.  But that weak battery will keep working and still slowly but surely recharge the capacitors and stay charged long after those video makers' cars will be dead in the water.  

Long story short, they are used in the automotive industry, and the people in those videos are simply spending money to make their cars inferior in many important ways by misusing supercapacitors in a way that is only detrimental.  They are not replacements for batteries because batteries store tons of energy, capacitors do not.  Used in tandem, however, they are a very good match and pick up the slack in the areas the other is weak.