They wouldn't need to run the ad business. Downsize and replace it with taxpayer dollars.
The reason to nationalize something existing in these spheres rather than build something new is because the network effects of these platforms make it near impossible for something competing to get a foothold. And if anyone could fail to compete against big tech, no one could fail better than the government.
I can't wait to see the level of grifts that will come when they really start to get desperate.