While I appreciate that Marx made a case, this is not data or evidence. It seems intuitively true, but that doesn’t really move you closer to real proof.
Essentially, competition forces prices lower, and automation and increased production lower the price floor. Automation is pursued because it temporarily allows you to outcompete, until other firms can produce at the same price, forcing prices to match at a new floor. This continues.
I’m not sure if you are trying to imply automation is a good or bad thing. Looking through history, the industrial revolution was bad for the workers of the time, but in the long run massively improved the living standards of everyone. Automation is a net good in my opinion. Competition is simply an accelerator, this is not really tied to the economic system being used. In capitalist or communist systems, firms that are protected from competition (by what ever means) do not innovate as fast or as effectively (see Intel as a great example of this).
Socialism is just the precursor to Communism.
While this can be true, it is not necessarily true.
I don’t see what birth rates have to do with anything.
As your population ages, the costs to care for them raise at an increasing rate. If you don’t have enough new workers to stabilize the economic base, the burden that an aging population places on the younger generation grows until it becomes untenable.