The jack supplied with the truck is intended for emergency use only - undoubtedly made as cheaply (cost savings) and lightly (weight savings) as practical, and as the OP noted, it is not even very efficiently packaged for that purpose - for any kind under vehicle repair work, an aftermarket floor jack is that much better by degrees.
For emergencies (read: tire changing at the side of the road in the middle of nowhere, with no phone service and nothing around) it seems to be barely up to the task, though it will work under most conditions likely to be encountered.
Having said that, it is my experience that if you get a flat, most people will call AAA in the States (or CAA here in Canada) and get the tow guy to change the tire, or tow it into a repair facility.
If you try to change your tires at the side of the road, there's another issue: whether you are able to access your spare tire.
As with most pickups, the spare tire carrier is prone to rusting out (usually the cable) or seizing up or getting jammed up with crud (the retractor gizmo) or other problems (e.g., low spare tire pressure).
Even though I lower and check my spare tire and carrier every year (at least), I've found that most of my problems with tires have been punctures, and the best way to tackle this is to carry a tire plug kit (to plug any puncture) and a portable air compressor to re-inflate the problem tire - don't even have to remove the tire in order to plug a leak, in most cases.
With this equipment on board, I've managed to handle (so far) any situation I've encountered where I otherwise would have had to use the jack/remove a tire.
Of course, this equipment doesn't help where I would have to change a tire - e.g., if a tire shreds or a sidewall punctures - but it's worked so far for me.
Also, it seems to be an unwritten rule that if you prepare for something to go wrong, you'll often be lucky enough not to encounter it.
Just my observation.