Some gyros can jump-start, as in pre-spin their rotor fast enough for vertical take off. It would add a bit of complexity but it'll still be relatively simple.
No, because the pre-spinning (torque applied to rotor) part happens on the ground with the rotor blades set to low/no pitch. The rotor then starts freewheeling (no torque applied or transferred from the rotor) and the blades' pitch angle is increased, which slows the rotor down, but momentarily creates enough lift to lift the autogyro into the air, where it immediately transitions to forward flight.
It’s still simpler and cheaper. The swashplate is much simpler. Much less power is fed to the main rotor as inertia is built up slowly and only for take off. The lower standard of reliability as worst that happens is you don’t take off. Engines are pushing a more forgiving higher speed prop, so engines and fuel are cheaper.
I don’t know how marginal jumpstarts are, possibly very, which would limit the utility. Especially where there is a runway nearby.
They also have some extremely dangerous flight characteristics.[0] To be fair, helicopters are even worse, but they require extensive training and are known to be dangerous.