Not true. An aircraft (yes, any aircraft, not just an autogyro) needs more thrust and less lift in order to climb. Let us compare a horizontal flight and a 45 degree climb. In a stationary horizontal flight, the lift equals the weight of the aircraft and the thrust equals the total drag of the aircraft. Remember that drag is always defined to be parallel to the airspeed and the lift is always perpendicular to the airspeed. Because it is a horizontal flight, the weight is perpendicular to the airspeed and can only be counteracted by the lift.
This situation changes in a steady climb. All aerodynamic forces have rotated 45 degrees, but the weight has not. Lift, drag and thrust have rotated 45 degrees clockwise while the weight is always pointed downwards. In this situation, the lift only has to counteract 70% of the weight (the component of the weight that is perpendicular to the airspeed) and also 70% of the weight (the component parallel to the airspeed) adds up to the drag. We see that the lift has decreased and the thrust has increased.
In climb, the carrying force is shifted from the rotor to the engine.
As a result, the rotor will loose rotational speed.
Myth: Autogyros are safer or unsafer than other aircraft
Isn't it cute?
Both stories exist. People who never saw an autogyro fly often think it is an aircraft that has come from a Disney movie. And obviously things from Disney movies can't really fly. But autogyros are not created by a cartoon artist and deserve to be taken more seriously. The fact that you haven't seen something before does not mean that it is dangerous.
Autogyros are also not safer than other aircraft, because there just isn't a fair comparison. In some ways it is safer, in some ways it is not. It is still an aircraft and if something goes wrong, it is a long way down. Please do not forget that autogyros form a separate class of aircraft with their own technical characteristics and behaviour. Alas there are stories of people who thought that they could fly an autogyro because they had either a fixed-wing or a helicopter licence. Now THAT is dangerous.
Do not be scared by the strict safety rules in aviation. Pilots generally do take safety very serious, and for a good reason: humans cannot fly. We are mentally and physically not created for flying. So we will have to live with flying machines and always be aware of the risks. That is the only way to keep aviation safe.
Myth: Autogyro theory is the same as helicopter theory
It's odd, but almost every helicopter textbook spends about half a page to the autogyro. In this tiny little chapter you can find a sentence like "The behaviour of the autogyro can be calculated with helicopter theory". You will never find an example of how that is done.
Let me put it clear: an autogyro is a completely different kind of machine. In theory one can build an autogyro around any helicopter rotor (not the other way around), but that does not imply that the machines are equal. An autogyro rotor is a special kind of wing. The lifting force is powered by the airstream that the rotor is in. That is why an autogyro needs a separate installation to develop thrust. A helicopter rotor is a special kind of propeller. The lifting force is powered by the engine and a helicopter does therefore not need to be in an airstream.
This really is a huge difference. An autogyro is a lift-generating aircraft, while a helicopter is a tilted-thrust aircraft. To understand an autogyro, you must first understand the "wing properties" of the rotor. You should be able to draw a polar curve. You should know how lift and drag connect. Only after you have found out how much drag comes with the desired amount of lift, you would know how much thrust is needed. Thrust to overcome drag, not to produce lift. To understand a helicopter, you must first understand the "propeller properties" of the rotor. you should know how much power is needed to generate the desired rotor thrust. Only after you hav found out how much rotor torque is generated, you would know how much power is needed for the tail rotor. Helicopters do not produce lift unless they are flying in autorotation.
Naturally, this is reflected in the theory. Helicopter theory is mainly about calculating powers and about the instability that comes with the rotor. Autogyro theory is mainly about calculating the aerodynamic forces and about the stability effects that come with the combination of the rotor and the engine.