this isn't a direct answer to your question, but here's aimixin on Marx and the elimination of private production (the first two paragraphs are setup):

Marxists do not claim people should just work for society because of some selfless feelings, Marx was personally annoyed with people who constantly said this and commented on it himself:

Communists do not oppose egoism...The Communists do not preach morality at all. They do not put to people the moral demand: love one another, do not be egoists, etc...the Communists by no means want to do away with the "private individual" for the sake of the "general", selfless man. That is a statement of the imagination.

—Marx, The German Ideology

The reason Marx saw a post-capitalist society as having socialized production, where people work for society, is because they have to. But, I know what you're thinking, "that's authoritarian!" But you'd be misunderstanding, he did not believe people would work socially because the government would tell them to at gunpoint or that owning a private business would be against the law.

No, he thought they would work socially because any other sort of economic arrangement would simply not be possible. Even if you changed the laws to allow for starting a private business, you still could not start one, because it would just not be something feasible people could do.

Why? Because Marx observed that in all capitalist societies, private enterprises always grow in scale, and the proportion of small businesses to big is continually shrinking. The more this goes on, the smaller the proportion of businesses owners to workers in a society becomes, the more and more small businesses go bankrupt and people the business owners then become regular workers.

Why does this happen? Because the government outlawed private businesses? No, because as businesses grow in size, the smaller businesses that can't keep up eventually just can't compete and are less efficient and go bankrupt.

Not only this, but as businesses get bigger, the barrier of entry constantly rises. Can you start a small business in your basement to compete with Samsung? Of course not, you need hundreds of billions of dollars in capital to even begin to compete!

Again, it's not the government making it illegal to own a business. It's the physical conditions of everyday life making it simply impossible to own one no matter what the laws say.

It is a misunderstanding of Marxism to think that what Marx had in mind was just to make all private businesses illegal. Rather, the vision he had was to nationalize the "big industry" which has already grown so large that there is hardly much competition anymore anyways, and then to use it to try and speed up economic development, because this will make more of the small business sector grow into big businesses, and then eventually they too can be nationalized.

Hence, Marx argued for a gradual, "by degree" nationalization process, alongside encouraging rapid economic development, "the development of the productive forces." Not just making all private enterprise illegal.

People would work for this big industry because there would simply be no other industry to work for and it would not be physically possible for them to start a small business even if the laws allowed them to.