Agreed. Anarchists often find US liberals incredibly frustrating to deal with, for some good reasons. A lot of that comes from them being one side of the status-quo ideology, in that both US liberals and US conservatives are descended from classical liberals. They’ve tended to resist scrutinizing most of their received wisdom because they largely haven’t needed to.
However, they’re currently more likely to be receptive to our ideas than they’ve ever been before. Some will end up being “go along to get along” Good Germans. Many others, possibly even a majority are somewhere between nervous and terrified about the future right now, and would welcome new ideas on how to deal with the situation.
They may not be ready to fully switch ideologies, but that’s something that depends on a more gradual background process. If we can refrain from anarcho-purism and meet people where they are, we can make a lot of progress and put ourselves in a much better position to survive and resist.
You shouldn’t come into an anarchist community and start answering questions about anarchism when you clearly haven’t done your homework.
But hey, since OP is interested in how anarchy would work, let’s go over how such a society would respond to the scenario that you’ve painted. Vigilante justice is never impossible in any society, but that doesn’t mean it would be tolerated. The requirement to have disputes arbitrated by a neutral third party is pretty universal. What differentiates anarchy is that arbiters are freely chosen by (possibly delegated) mutual agreement, instead of the state forcibly inserting itself into every dispute as the supreme arbiter.
Let’s say the thief was a member of a commune. Since the thief is dead, their dispute with the baker and the lynch mob can be claimed by their next of kin, or closest equivalent. Either way, we’ll say that the dispute gets delegated to the commune as a whole, which collectively handles security and dispute resolution for its members.
The baker has a contract with a company (probably organized as a workers cooperative) that offers security and dispute resolution services. For simplicity, let’s say that the members of the lynch mob also use this company’s services.
The commune and the company might have different sets of rules that their members agree to, but it’s reasonable to assume that they both recognize:
From there it’s just a matter of negotiating what restitution is owed to whom. Perhaps the commune and the company can’t come to an agreement on what exactly is owed, so they agree to defer to a neutral arbiter of their own. They may both be members of a local federation of dispute resolution bodies, which would simplify handling this.