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

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That happens anyway. The presence of USPS just made a duopoly a tripoly 

Can’t think of a sector where govt presence creates large n supplier markers

If there’s no state offering sufficiently to exert strong downward price pressure…private sector will cartelize