When tuning FF cars with racing tyres I found out it could be useful for some of them to use racing softs in the front and racing mediums or even hards in the rear. This would make the car looser, getting rid of much of the understeer, and also much more interesting to drive, behaving in some ways like a MR car.
Technically, that doesnt remove understeer (front wheels losing traction), that just adds oversteer (rear wheels losing traction), which FF cars normally dont have. You can get a similar effect without using different rear tires by shifting the brake balance to the rear, so the rear wheels run out of traction sooner. Brake into a turn and the back should slide out, kinda like a RWD powerslide, except you maintain it with the brakes rather than the throttle, and you can get out of it more easily by laying down power (because this doesnt rely on the power wheels losing traction), and the fronts will pull the car back into line, so it transitions smoothly from sliding to driving.
However, theres more to cornering than just rotating the car, you also have to overcome momentum. If you do one without the other, youll just slam sideways into the outer wall. If the car wont rotate, thats just a symptom; if you manage momentum correctly, the rotation will reflect that.
That said: In a FWD, you dont
have to rotate (as much) in order to lay down power toward the turn, because the power wheels are steerable. If youre coasting through a turn at the limit of traction, add a touch of power and it will actually turn faster, because you can add power and thus momentum
in the direction of the turn. (In RWD, you can only lay down power in the direction the nose is facing, which contributes to the very momentum that the (unpowered) steer wheels are trying to overcome in the first place.) This gives you access to a part of the traction circle you cant access by unpowered steering alone, and this is why FWD actually has
less understeer than RWD in experienced hands.
(The same can be said of 4WD, to some extent.)
Add a touch of braking thus, using all three inputs at once! to slow down
and shift weight (and traction) onto the steerable power wheels, and boring old FWD can become quite acrobatic!

(Especially
in Rally driving, but also useful as an emergency/corrective measure on tarmac.)
This all stems from car control and things like suspension/brake setup, without needing (nor wanting) different kinds of tires in front and back, which brings me back to my original question.
Indeed, with weight transfer management, you can adjust the front/rear traction balance pretty much at will, not just in the pits. With different tires on each end, it means you
have to do this just to get equal balance when thats called for, such as coasting through a turn, which itself presents little opportunity to shift weight around.
The only different-tires context (that I can think of) that doesnt involve simply which end gets more traction, is dry vs rain tires. I suppose, in this case, you could make one end gripper in dry conditions and the other end gripper in wet conditions. But is there any point to that? The inconsistent behavior would be far worse than the loss of traction as rain accumulates on the road (or evaporates from it).