What purpose does changing to braided hoses, polyurethane bushings and the sort do to improve performance and reliability in general
Braided hoses expand less than all-rubber ones, because metal which surrounds the braided hose expands less than rubber.
When applied to brake hoses, this makes a difference in high performance driving where high brake pedal pressures are involved - due to reduced expansion of the lines, brake pedal feels more linear and consistent. This is especially important in cars without ABS (anti-lock brakes), which are quite common in racing to this day, where the driver must manually obtain maximum braking power from the vehicle without locking up the tires.
When applied to fuel/oil/coolant hoses, the purpose can be safety. Again this is more common in race cars, sometimes these fluid lines run through the cabin of the vehicle and some regulations require fluid lines going through the cabin to be metal braided, to increase puncture resistance or reduce likelihood of bursting due to unexpected high pressure, as for example could happen in a fire.
Silicone coolant hoses expand less than rubber ones but these are more for the bling than for a functional purpose, though they can be installed for longevity purposes as rubber hoses do wear out over time.
Polyurethane bushings deflect less than rubber ones. This gives the car higher resistance to body roll, meaning the body stops rolling sooner and oscillates less. This improves responsiveness of the vehicle when it is taken through a sequence of turns in alternating directions like a chicane. Polyurethane bushings also transmit more of surface imperfections to the chassis, resulting in a firmer ride in general (as well as making the driver more aware of what surface they are traversing, i.e. bumps/dips).
When it comes to street cars, OEM components usually work just fine but over time do need to be replaced. Race cars that have aftermarket components generally have much less mileage on these components than street cars have on OEM parts.
Reliability comes from servicing and preventative maintenance.
Poly bushes can be noisier for some people- they are not as compliant usually as the original parts. The original spec parts cal last 100k or more so poly bushes are often fitted as a “tune up”...
Most brake hoses are braided, but if you mean the ones with stainless braiding on the outside then some fit them for looks, others to have a slightly harder pedal.