Have you taken the car completely apart and checked every bearing? Just because the car has been carefully driven and never crashed does not mean you do not have a bad bearing or binding somewhere in the drive line. I would personally take the entire car apart and inspect every bearing, hinge pin, shock, suspension arm, suspension links for damage and make sure all these parts go back together as they should.

You seem pretty hell bent on changing the toe even though many people with years of experience here, not to mention the engineers at HPI, all agree on the need for rear toe. On my Tamiya TT02 the stock hubs have 0* of toe and the car was a total handful until I installed hubs that have 2* of toe in, it made a world of difference in the stability of the chassis.

I too run on unprepared tarmac, actually tennis courts, but they are always dusty and very slippery. In the 6 months I have had the car I have used 5 different sets of tires trying to find something that will provide a bit more traction. My current favorites are the Ride 26MM Radials and the Tamiya Super Grip Radials. They both offer a bit of tread and a pretty sticky compound for this sort of use.....the Tamiya's seem a bit stickier but seem to be wearing quicker than the Rides.

Before you start reinventing the wheel I would tear down the car and look for any issues and carefully reassemble it make sure everything is even side to side. I would then try new tires. Are the tires you are using original to the kit? How old are they? Even through the tires might have plenty of rubber left on them does not mean they are not dry and hard and that can start to happen pretty quickly depending on where and how they were stored. I can tell you there is a reason why every competitive chassis on the market has toe-in built into the chassis.

Cheers,
Jim