By left anarchism, I assume you are referring to pro-democracy socialist anarchism as opposed to anti-democracy capitalist "anarchism" ("anarcho"-capitalism.) I should note that anarchism is an ideological movement with historically left-wing tendencies, and on this site the term "left anarchism" is redundant.
There are a few things to bear in mind when analyzing democracy:
1. Humans are generally objective, moral people. Only a select few are selfish & greedy; not nearly enough to push a democracy over its tipping point.
2. If left anarchism were ever truly achieved, people would operate in a socialist mindset; each would have access to enough resources to "opt out" (leave) a collective if they ever felt unsatisfied.
3. One decision must be carried out eventually, and I'd rather it be that of the majority that that of the minority. Even Natural Law is maintained through coercion, as those who try to defect are put back into place through the threat of violence. If natural law were to be carried out solely through the consent of the minority, it would be a case of minority rule, an even worse situation than majority rule.