Adding names to every car would not only (likely) breach several trademarks (LCS already does this with guns, but to a small extent), but also unnecessarily complicate the game. It's why I think we don't need more guns. Adding more cars when they're not meaningful would make the game mechanics less obvious and also create a doctrine in the player of 'collecting them all'. I think the number of car types right now could be cut down on, really. A good example is SUVs, which are super easy to steal, sell for a good chunk of cash and don't let you get caught easily.
As for the rest of the proposals, they're pretty solid.
I have to disagree with you on that last part. Technically, the game would play exactly the same even if only one type of generic "car" were available - even the year and color are unnecessary details. But while the game doesn't necessarily NEED any different vehicles, they do add some flavor to the game - even if it's only useful for roleplaying.
In fact, I'd be thrilled to see more vehicles added to the game!
There is a danger to adding more though. Too many options for the player and you start making the game worse rather than better, because players will become indecisive and confused.
The current car choice is pretty easy to navigate because people tend to know what different car types are, and they can guess at some they may not know. Like, I think "station wagon" is an option? That's a term that is rarely used for modern cars in the US anymore, but it's not going to run anyone's play experience that it's in. Even if you don't know what a station wagon is, you don't need to; you'll just assume it's a coup or a truck or a buggy or something, and it won't make your play any worse for it. It's a choice and people probably rarely pick it, but doesn't make the car menu notably harder to deal with by its existence.
If the player has to choose between five types of sports cars though, you're going to run into an issue where players aren't sure which one they
really want -- they know they want a fast getaway car, but have no idea whether the Firebird or the Racer is better at this, and feel frustrated before picking randomly and wondering if they got it wrong. This is a distraction that makes the game less fun and makes players feel slightly helpless. So you need to temper the variations to ensure the game remains intuitive and empowering, rather than causing confusion.
Having cosmetic differences that apply
only after you get into the car theft minigame would avoid some of this trouble, as long as those differences really are just cosmetic. So I could see something like, you pick sports car, and it says:
You're in luck. You find an unattended silver Firebird, a high-performance 2015 Japanese import.Now you can get variation, flavor, and detail, but without muddling the game with too many choices. The player picked sports car, and the game says they got what they wanted, plus more details to get their imagination going. Average players won't grind for specific cars because they don't know what sort of car is better, and expert players who DO know all the mechanics won't get bored grinding either, since they'll know these are genuinely cosmetic details under the hood.