I have to say... this should be the template for how feedback is given... very nicely done sir.
As for spooling and unspooling of the Jets to quickly, that can actually be remedied.
If you imagine these rigs as crazy hybrids that use both a jet engine and piston engines as forms of propulsion... then it becomes easier to understand how this might work. As noted... when you come off the gas and jump back onto the gas, the propulsion from the wheels and jets seems to be almost instant... We also know that while piston engines can change their speeds rather readily, turbine engines take a while to spool and get up to speed. So... how can we bypass that issue?
Well... if there were some form of clutch system (engaging and disengaging) and a step up box (just a geared step up to sync the RPM's), we could use the crankshaft from the piston engine and connect it directly to the prop shaft of a turbine engine... simply engage the clutch, rev the piston engine, and within a second or two, you are up to full RPM's on your Jet engine ready to be dishing out full power.
On the flip side, during gear changes, since the Jet will already be at full RPM and power, when the piston engine changes gears and loses RPM's, the Jet engine's clutch could theoretically re-engage and somewhat help pull the Piston engine back up to RPM giving a boost of low end grunt to the wheels in every gear (except first gear because that is where the piston engine would be spooling the turbine in the jet engine). For the upshifting power boost to work, it would also need a torque converter that allows for some slip in the system as well so that the Jet wouldn't loose to much RPM's when the clutch was re-engaged.
This little technique could also be used on the startline as well for first gear... in the old rollcage games you used to be rewarded for nailing the throttle at the perfect moment at the start of a race and would get a big boost. Well in this instance, you would need to hold in your throttle to build your turbine RPM's to their optimal launching RPM's... granted, if you rev the turbines too far, then you will have too much power to engage the system on launch, so to keep from damaging your car, it will wait till you have built up speed with only the piston engine and the turbine thrust first before it can re-engage the clutch and give you that boost. This could make the starts of races rather interesting, as rigs with bigger turbines would need the throttle held in for longer to be able to achieve their optimal launch RPM while smaller, lighter rigs with smaller, lighter turbines would need much less time to spool up... meaning that it's much easier to mess up the start if you jump from rig to rig rather often.
As for startup sequences at the start of the races... I think that would be epic... hearing the big piston engine sputtering to life... with possibly a bit of black smoke, and then the drain on the engine as it spins the Jets to life and each one pops to life individually... oh man... that would make for such an epic start of every race.