Jump starts have always been a crappy approach IMO. I think a small hand-cranked generator into a decently-sized supercapacitor could be a much better way to kick over a motor. I feel like you could get a reasonable jump start with just a minute or so of cranking. Certainly faster than waiting for someone to find their jumper cables or drive out to assist you.
Keep in mind that this is already sort of a thing for cars with manual transmissions. In this case, the proposed supercapacitor is replaced by the kinetic energy you are pushing into the car.
According to a google an average car might take ~9000 joules to start. Also according to google, an average hand crank only puts out about 15 watts. So you might be cranking for at least 10 minutes.
If you could convert gravitational potential energy into charged battery energy at 50% efficiency, you would need to raise something weighing 2000 lbs to a height of 6'6". Your own car maybe? You could have tripod legs that deploy from the side of your car, allowing you to winch it up that amount, and then to start it you just drop the car.
You've really only got to fire one plug one time to start a car that's in a fundamentally startable condition. A battery that doesn't have enough juice to turn a starter motor (which requires a lot of amps) often still has more than enough to fire the plugs if you can turn the engine over some other way.
Several times in my life I've had to roll-start a (manual) car with some regularity[0]. You don't have to getting it going to even a brisk walking pace if it's a small (< 2 liter) four cylinder engine. If you clutch out and back in pretty quickly, it'll start if it'll fire. If the battery is too dead to run the ECU and/or plugs, you're pretty well dicked unless you can find somebody to jump you.
Now, neither of these applies to an F-150, since you can't get one with either a manual or a four-banger, but practical experience indicates that for at least some cars, it's got to take less than 9000 joules to start the engine.
In rough numbers, if you have a 1500 kg car moving 1 m/sec, you've only got 750 joules to work with. In all cases, I wasn't trying to do this in the dead of winter.
[0] On 2 different cars:
1) I left the headlights on two days in a row and discharged the battery pretty deeply. I got it back after a couple days of driving my short commute.
2) I had an intermittently bad connection between the battery lug and the wire running to the starter motor. I finally figured it out after it did it at night and I could see the sparks with the hood open. That was after 2 or 3 months of bump-starting the car at least once a week :-)
4-cyl engines are easy enough to start with a hand crank off the crankshaft. Standard starting procedure for a long time, really. My first car had one, and it was made in 1963.
Won't work on the hulking mammoth engines of a modern F-150, but maybe a step down ratio maybe 1:4 (and a bicycle) and it could work.
Or a small lithium battery to supply the necessary current which is already a thing you can buy on Amazon for $80 and will start any vehicle short of an excavator.
To agree with this some, the super-capacitor jump starters should definitely change the necessity of jump starting, as they provide more amps closer than 6' cables.
Unfortunately, education is lacking; my father is an example that kept jump starting even when he had a super capacitor jumper.
Keep in mind that this is already sort of a thing for cars with manual transmissions. In this case, the proposed supercapacitor is replaced by the kinetic energy you are pushing into the car.