Not exactly,
First the extremely high heat and pressure that develops causes a rapid, explosive expansion. Then as expansion progresses and the gas rapidly cools, then the cold regenerator moves into place, at a 90° offset, same as a standard Stirling engine.
Obviously, you still do not understand and probably never will.The problem with that is the cold gas is spread out over the entire cylinder and is away from the cold plate. The only time the entire gas is next to the cold plate is when it is compressed and hot.
The "entire gas" next to the cold plate "when it is compressed and hot" never happens, as already explained.
I'm sorry, though conceptually relatively simple, this is all apparently way over your head, as is the operation of a Stirling engine. You're incapable of escaping your 1820's Carnot "Law" indoctrination.
A regenerator does not absorb cold. There is no need. Cold does not exist. It is merely the absence of heat. You can't collect cold in a regenerator by "waving" the regenerator through the cold gas.Maybe if you had three plates, hot, insulated, cold, then waved the cold one through the cold expanded gas.
It would generate power output the same as any normal Stirling engine. It is functionally identical to a normal, standard Stirling engine.Your description will produce zero power. It will require power input, just like a heat pump. And any generated heat differential will be trapped inside the machine. I assume you know that.