In the interest of trying to contribute something helpful, and apologies if I'm telling you plenty you already know, but the suspension setup vs tyre can make a big difference and may be related to the snappy oversteer that you're experiencing?
Obviously we're all aware that slicks have a much narrower window of slip before they loose grip sharply, but you can end up with some really nasty snappy handling characteristics (that are manageable with a controller but awful with a wheel) if the suspension setup isn't changed to match the tyres.
In simplistic terms, you can't ask as much from the less grippy tyres, and so you need to go softer with the suspension, else as soon as you begin to load the tyre up, it falls off a cliff as you're asking too much of it.
The comfort tyres need more 'road' style suspension settings, as a rough guide 1.5 - 1.8Hz on the springs, 4 or 5 on the roll bars, and a max camber of about 2 degrees or so.
The sports tyres can take a bit more spring rate (around 2Hz) and maybe half a degree or so more camber.
Slicks you can then push on up to say 4 degrees of camber, springs rate 2.5 - 2.8 (harder if you have a lot of downforce), 8 or 9 on the roll bars.
I thought it worth mentioning because you can have a really nice handling road car on comfort softs, you can fit the street or sports suspension and it sharpens it up nicely but stays pleasant to drive, and then you fit the fully customisable suspension and it defaults to loads of camber, excessive spring rate etc., and suddenly you have a snappy mess on the same tyres...
I didn't really think the suspension settings in GT7 were that significant until recently, a friend and I were playing and as he is a chassis engineer for a major car manufacturer and has been a race engineer in the past, he actually knows what he's talking about when it comes to suspension. He watched me do half a lap in something, tweaked a few settings and totally transformed the car into something wonderful to drive. It blew my mind
I've since applied what I've learnt from him in tuning my cars and even with my basic grasp it makes a really noticeable difference to how the car drives and also how it lets go at the limit.
Naturally it makes no difference to that ridiculous infinite wheel spin behaviour, but you really need to be a gear higher than necessary and still have enough power to blow the tyres loose for that to take effect, it really seems to be a wheel speed vs vehicle speed issue, once there's too much of a difference between the two, it's like it just detaches the wheels from the drivetrain!