Many on the right also conflate markets with capitalism. It's a lot easier to paint anti-capitalists are loons if you portray them as being opposed to markets.

Amen. Markets are what really makes the economy function, who exactly owns the capital is probably not all that important.

Not to mention how "Market" is a collection of behaviors which occur naturally in an economic environment absent direct intervention.

It is abjectly impossible to intervene in an entire economy, so the way to handle markets is to understand how they work and apply intervention as is actually needed, like say for example establishing standard setter state runs in industries where competition has turned into a race to the bottom to force better behavior like federally run hospitals or grocery stores, or establishing a more ethical alternative to industries that are a net negative to public interests, like building out short and mid hop high speed rail to undercut the commercial viability of short and midhop flights.