I don't find the Bidgestones too bad to be honest. They squirm about a bit in the wet but they always tell you what they are doing.
There are a lot of Bridgestones.
S-02 on the early cars would have killed you in the wet.
RE050MZ came on the later ones which were better, these are now replaced with RE050A which are much better in the wet (still not great). The S001 are a step change again in wet performance.
All in my experience, also T1R toyos ruined the car as the sidewalls were too soft. The rear end felt disconnected in combination with the Passive rear steering.
Im trying som F1AS2 next. Honda's need stiff tyres so this could be a gamble however a bit of compliance would be nice as I run the Koni's on stiffest and have Polybushes all round with 1.5deg front and 2deg rear camber
The passive rear steering is what gave the car its reputation (and high insurance). The combination of nervous drivers lifting off mid bend, rear steering tucking the nose in, weight transfer off the rear wheels, stiffer rear suspension and a rearward neutral point and non-progressive, crap in wet, Bridgestone S-02s were a recipe for disaster.
The 2004 on have softer suspension, more understeer dialled in on the limit and different toe control pick ups on the rear subframe to make them 'softer' and arguable a less involving drive. However it must be said, the early ones can be quite nervous.
The electric power steering and the fact you sit on the rear wheels does numb the steering feel, but a front lower strut brace helps to bring that back and tighten thinks up. Replacing the rear lowe tin foil brace with a stronger one is also a well recommended modification.
The standard brakes are really good, a master cylinder brace helps remove the small amount of flex in the bulkhead.