The general advantage of auto-gyros is slower speed of flight than fixed wing aircraft while having similar operating costs due to simpler mechanical design compared to a helicopter.
They do however still require a runway, and 'operating cost' is not necessarily a great measuring stick for hardware that must be shipped to Mars before use where total weight and performance matter. They are also constrained in forward speed in the same way as helicopters.
The runway length is tricky to estimate, most fixed wing designs math to needing take off speeds near mach one, and therefore indefeasibly long runways even by earth standards and a pure autogyro is not going to improve on that much.
One key item in the question is forward thrust. A jet engine will not work on Mars since the atmosphere is already carbon dioxide. Pretty much any feasible aircraft on Mars will depend on electric propulsion, and the physics mean that the largest possible blade length is needed.
It is much easier to mount very large propellers horizontal as a helicopter than vertical as conventional aircraft, leading to the choices made with Ingenuity, and a hypothetical autogryo needing very long landing gear or some sort of multi engine setup on a wing, at which point making it a fixed wing Osprey style tilt rotor starts making more sense anyway.
None of this makes an autogyro a good solution for current flying on Mars, at least until the capability to build large runways exist. It is possible that with a human population on Mars but limited supply from Earth autogyros might be a local constructed solution for flight being mechanically simpler than full helicopters, using catapult launch and the ability of the rotors to store kinetic energy to allow 'short' landing on a runway that needs to merely be long rather than a feature visible from Earth.