A game where the primary action and challenge consists of jumping skillfully over platforms. Jak II and (especially) Jak 3 are packed with shooting, "driving" and further kinds of action sequences make up the majority of the game and in particular, the majority of the challenge. The platforming which undoubtedly is (a relatively small) part of the game is trivial and certainly not the main challenge in the game. Jak II and Jak 3 are about as much of a platformer as Banjo Nuts & Bolts is.
The non-platforming elements you describe are present to a large degree in almost every game listed in this thread. Lets examine Jak 1 as an example. The gameplay takes place across 9 large main levels along with 7 smaller areas (1 tutorial area, 3 hub areas and 3 transition areas between hubs). One of the main levels, the Precursor Basin, consists entirely of "driving". (I love how you put that in quotation marks by the way... are you Donald Trump?) The "driving" elements also make up a fairly lengthy section of the Misty Island level. Throw in the three smaller transition levels between hubs, which also consist entirely of "driving", and you'll get that approximately 20 percent of the game's precursor orbs and power cells are collected whilst "driving".
Jak 1 also features heavy combat elements which come into play during every main level in the game, often placing you in closed off areas and forcing you to fight off a large number of enemies in order to collect a power cell or to continue your way through (see the arena in Misty Island, the ambush in Boggy Swamp, the lurker infested cave in Snowy Mountain, the spider tunnel in the Spider Cave...). The game also features shooting elements via yellow eco powerups which play a large part in a few levels (in particular the Boggy Swamp and Spider Caves along with the game's two boss fights), minigame missions in Forbidden Jungle and Boggy Swamp (which by the way are complete shite), as well as extremely simplistic traversal through the tutorial level, the three hub levels, and the entire first main level in the game (Sentinel Beach), which can be in no way described as "skillful jumping over platforms". Even the tutorial mission in Jak 2 requires more platforming skill than traversing these areas in Jak 1.
To summarize, if we judge Jak 1 by your own standards, calling it a platformer would also be a huge stretch. This would also be the case if we did a similar analysis of most other games mentioned in the thread. Unless you want to throw out all of those games from discussion, you can't exclude games simply for containing elements unrelated to platforming. Even with all of the other elements it contains, Jak 2 still consists mainly of platforming missions similar to the ones I linked in my previous post, so anybody familiar with it can easily see that it's a platformer. But hey, I've only completed Jak 2 around 15 times so maybe you're more qualified to talk about it than I am.
I'm not too familiar with Banjo Nuts & Bolts, by the way. Can you point out any areas in the game that are filled with platforming changes similar to the five missions from the first hour of Jak 2 that I linked in my last post?