That’s not what a class is, certainly not in the Marxist sense. Classes are social relations to production and distribution, relating to ownership. Managers are working class, as are administrators within socialism. If you want to discuss the relationship between subsections of the working classes, then this is a different discussion, but class isn’t simply any way you can identify a group. Teachers are not owners, nor are managers, nor are administrators, when their relations of ownership are the exact same. Organizing at scale creates managerial and administrative positions out of necessity.

Publicly owned and run press, as exists in China, for example, is socialist press. These are worker-run and managed. I think you may be hinting at cooperative, horizontalist management and ownership, but this isn’t really socialist in the Marxist sense at all. Scientific socialism is about the collectivization of production and distribution in one unified system, democratically run in the interests of the working classes. Horizontalist cooperatives fragment the working classes and pit them against each other, and are easy to take advantage of by foreign powers (as the west has done many times already).