You can’t have Capitalist elements like, commodity production, wage-labor, money, markets, and call that “socialism”. All of the inherent contradictions and harms of capitalism remain (because its capitalism), like theft of surplus value, imperialism, and alienation.
The term leftist is considered to come from the fact that people sitting on the “Left” in the Estates General in France. Generally, liberals, (Jacobins, Third Estate). There is very poor sourcing for this online, but Wikipedia cites some untranslated difficult to source french book.
In mainstream discourse, “left” takes on a number of meanings, but to the extent that left is meant to mean anti-capitalist, and/or, Marxist, anarchism has left wing currents, hence, left wing anarchism.
I find people are less uncomfortable when you mention Market Socialism, and show them how it’s not too radical.
Cool theory. Still Capitalism (and still creates all the same problem capitalism does).
There are definitely “right-wing” anarchist currents that are popular and not as unhinged as how “anarcho-capitalism” is generally characterized, as in, anarchism that espouses the use of capitalism, but I have not really seen any genuinely “socially conservative”, anarchists. Pacifism is reactionary and emboldens the status quo.