You cannot call it a free market if government lobbying exists and is the primary tool for establishing monopolies.

Nor you cannot call it a free market if the government is allowed to favour or penalise one entity or one sector over the other.

I know words don't mean anything these days, but there is no angle in which you can call modern economy a free one.

“a real free market economy just hasn’t been tried” where have I heard that before?

As soon as we admit that humans are easily corrupted, the sooner we can stop finger pointing and start developing systems to mitigate the natural tendency toward corruption.

Isn’t that free market capitalism? We assume everyone will act in their own best interest and that this results in the most efficient allocation of resources.

That’s just one way of becoming a monopoly.

Is it a free market if there’s antimonopoly agency?

Government lobbying is just one way to put a hand on the scale of the market. Legislation, incentives favouring unprofitable endeavours, etc.