It won't increase it that much and its already going on for a couple of MHz wide which is quite good considering the efficiency.
Taking a bog standard vertical, using 2mm diameter element you're going to have a bandwidth of 900kHz < 1.5:1
Increasing that to 10mm you're looking at around 1.1MHz <1.5:1, just a 200kHz difference.
Now that itself is maybe worthwhile if you're building a base station antenna but for a mobile antenna given the problems it causes then the gain in the bandwidth isn't worth it. It would be far heavier, pretty much inflexible and the mounting solution would be a lot more complicated.
Unfortunately part of how strong a magnet sticks to something is how thick the metal is that it is sticking to. When you see them being demonstrated in stores and at hamfests they're typically attached to a thick steel plate which is a lot thicker than the roof of a car and certainly a hell of a lot thicker than the roof of a modern car.
I know for certain that a Sirio 145 mount with a Yaesu ATAS120 on falls off the roof of a 2004 Chrysler Grand Voyager at 70MPH. Been there, done that.
There's various definitions of "work", many of which are no more than "do I have low SWR and can I make contacts" which doesn't tell the whole story. You might be making contacts but are you making as many as you could be and do you get a lot of noise you needn't do because of common mode and ground issues?
Using one of those toolboxes you typically find in the bed of an American truck is probably going to turn out OK other than the body of the cab masking the antenna possibly causing some matching issues.