Your presumptive understanding of what constitutes "the Left" confuses literal (political science? economic?) definitions, which exist outside history and the development/evolution of ideas, for an overarching reality.
The question has much more to do with lived history, the evolution of ideas, the changing relationships of ideologies with (state) power, and the fluid use of language. Semantics is only a small part of this complex equation.
If you have a strictly binary worldview, then anything not strictly bound by the relationships of production/property/profit that we understand as capitalism will be characterized as "socialism" -- which you equate with "the Left." This is very simpleminded. Some understand socialism as meaning that the producers determine how production and distribution is to be accomplished; some understand socialism as meaning that agents of the state (through the elected representatives of the majority of producers) determine how production and distribution is to be accomplished; some understand socialism as meaning that unelected experts/bureaucrats determine how production and distribution is to be accomplished. Aside from the unexamined question of why we would want a system that coordinates production and distribution, none of those has anything necessarily to do with the abolition of the state -- in fact the latter two require a state. Socialism, defined broadly (and vaguely) as the opposite of capitalism, is definitely not incompatible with the existence of the state. More importantly, those proudly self-defined socialists who have had governmental responsibilities in various countries since the 19th century, have increased government intrusion into the lives of the average producer. In addition, most of them have reveled in the suppression of anarchists and other authentic revolutionaries. The worst example of this was probably the German Social Democrats who were in uncontested power from 1918-1923/24.
In terms of socialism's relationship to capitalism, that's a more tricky subject. Naturally it depends on how one defines capitalism. If we take it to mean an economic system that is predicated on the existence of private property for profit, then we need to look at what private property is, and how it is maintained and defended. We need to examine what profit is and how it is generated. We need to look at how a medium of exchange (usually money) is monopolized and enforced as the only legitimate medium of exchange -- by the state! If the state, with a socialist hue, decides to interfere with the market (itself a convenient fiction of autonomous chaos), does that make the market less capitalist? Maybe less profit-driven, maybe with less profit for particular property owners, but it's still capitalist as far as anyone can tell. So the co-existence of socialism with capitalism is perfectly possible, and has occurred throughout history.
If anarchists are interested in an irreversible abolition of the relationships of property/production/profit as maintained and defended by the state, and are interested in an irreversible abolition of the state as the primary location of institutionalized domination and hierarchy, then it makes sense that coherent anarchists would not want to be considered to be on the same side as anyone who is the least bit sympathetic to aspects of capital and the state. Non-anarchist anti-capitalist and anti-state radicals who reject the official Left are clear that the Left is the the left-wing of capital. It's high time that anarchists recognize this too.