My car broke down on my way to the airport, the tow truck took forever to come tow my car home and I missed my flight. Who is responsible?

It seems you are responsible for it. You should have maintained your car better, or hired a taxi to get you there on time. 

Ultimately, you are.  The airline is not required, obligated or otherwise responsible for you missing the flight.  That also means, no refund unless you bought a full price unrestricted ticket.  

Unfortunately it is probably you. I know train companies aren't responsible for consequential loss when they run late. Won't the airline let you reschedule? There'll be a fee it's but not that much

This question is ridiculous. Only you are at fault

Sorry , you are . It's not the tow truck drivers fault your car broke down . He was probably busy on another call .

S#@t happens! Welcome to the club

Unfortunately, it's your fault. Your car, your insurance, your journey to the airport.

Yea it ain’t like the tow company had anyone else to tow that day. They got your call and everyone screamed stop and they all dashed to see who could get to you first.  

Who should be responsible, the person who owns and drives  the car.

Who is responsible, the person who owns and drives the car. 

That's on you.  you could have left earlier, or maintained your car better, etc.  The tow truck got there when they got there.  It isn't the tow truck driver's responsibility to make sure you got to the airport on time, it's their responsibility to tow your car. 

It seems you are responsible for it. You should have maintained your car better, or hired a taxi to get you there on time. Tow trucks are often busy and take a while to respond. They have their own schedules to keep, and yours is not their problem.