Is there a name for the type of government where the government pursues consumer protection, antitrust, etc policy by directly running one competitor in every important industry, instead of making rules for every participant?

Why would you expect this to work? State-run enterprises are not exactly great vehicles for creating competitive pressure

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So subsidize the state offering sufficiently to exert strong downward price pressure… this means you’re dampening competition not promoting it

Increase the pressure enough and the private sector become unviable

Depends how big the market is as well. Here in lil'ol Aotearoa pretty much every major vertical is dominated by a two player oligopoly, with the rest left to fight over the crumbs.

They create market distortions distortions and get unfair regulatory and fiscal treatment, but I think they can absolutely create desirable competitive pressure. Case in point: BSNL vs private telcos in India.