Not all airplanes. Champ, J3 Cub etc.
May be your assumptions are restricted because of the airplanes you fly.
Regardless it is still important to understand why you are doing the technique and not just learn by rote.
In gyroplanes, you may be climbing at 60 knots and you may want to do your downwind at 60 knots also and the same is true in a 150. You climb at 67 on crosswind and you would be leveling out on downwind right around the same speed.
A generic prescription of levelling by pushing stick forward always does not apply to even all scenarios in an airplane. Not every flight and every climb is a cross country. You are supposed to cut power and adjust attitude simultaneously in an airplane if you want to maintain airspeed. It works. It also works the same in a gyroplane and works the same in a trike.
The difference in a gyroplane is even if you wanted to level out at a faster speed than your climb speed (Vy), you would still reduce power, level and then smoothly increase speed in a coordinated way with power. The reasons for that are obvious. Shoving stick forward with a bunch of power on is not a good practice in a machine that does not like to be unloaded. If you do it smoothly enough, nothing happens. If you do it like you are flying a 182, you'd see the G's go down for a little bit. Still likely nothing will happen in a gyroplane with an effective HS, but it is just not a desired practice in a gyroplane.