"- New tires?"
..............Yes,you should get new tires.
I'd definitely file the center ridge off the rims and put some mini-Z tires on it
if you want to decrease lap time(you wrote "increase track time" but I'm assuming you meant the opposite)
What kind of track are you running on anyways?
The type of surface you run on should determine the type and degree of tire.
As a rule of thumb you should always run a softer tire in the rear.
I'd say tires can account for at least 60%(probably more) for handling. The right tires will make a world of difference on any car.
So I'd start with tires, then mess with the suspension and other stuff after.
Buy a few different sets to tune your car to the type of track/surface(if you can afford it)
I'd try and save some weight from the body rather than the chassis.
The body has more excess weight plus you'll lower the center of gravity.(and decrease roll-overs)
I shaved allot of weight off of this normally heavy body........
That body is now lighter than my mini-0Z 05 NSX body.(I could have even gotten it lighter than it is now)
I know it's a little different than the supra, but you get the Idea.
Also, one of the best things I've done to a gen-1 was install a micro-T board and servo. I've seen used ones go for as little as $10 delivered for both on some forums.
That would definitely fix you steering problem as well as greatly improve you handling, so it's worth keeping an eye out for a cheap one.
Also the biggest challenge with the gen 1 xmod is the rear suspension in my opinion, but it sounds like a fun project for you to tinker with anyways.
Good luck, and have fun.