So I spent some time in anarchist circles, and what you've just described comes up regularly. It can actually be even worse than what you just described, specifically because the informal managers are also not given the social status a formal manager is. That means that in worst case scenarios and poorly working teams, they are expected to both keep up the workload of the position they officially hold, while also keeping up the managerial position (burnout), or alternatively they are regularly challenged as to the validity of their managerial knowledge.

That said, in rapidly changing situations, that second component is actually a great strength, as it ensures a free market of ideas within the potential "manager" pool, and allows an organization extremely rapid ability to respond to change. And so long as there is some intra-team awareness and team focus on measurable goals, the former can be placated as well.

There is another weakness, in situations where the managerial overhead is extremely intensive, unchanging and time consuming, ie, when the situation results in people who informally spend all of their time doing management, simply because it is more efficient for one person to do it, than that burden to be shared on everyone. The positive to this is because the position is informal, the number of people working on it will be exactly proportional to the amount of work required to be done, while in a normal organization there is a drive to increase management as much as is sustainable by the organization, both as employees attempt to gain higher salaries, and individuals already in management attempt to retain theirs. The negative is that as the informal position moves between individuals willing to put in the time to do the work, if there is no process to track who has done what, accountability can be an issue.

To attack this accountability issue, the traditional anarchist response has always been to formally recognize these positions, and work hard to ensure that these people are replaced regularly by democratic process. Someone below linked the tyranny of structure-less management article, which goes into far more depth, and arrives at the previous sentence as its main conclusion.

In our modern software startups, these are exactly the situations we attempt to automate, so it is less of an issue.

So now we are left with an issue of the size of the organization which wants to attempt this managerial structure.

In small groups, such stringent processes as just described need not be defined. If you have a decently working team, this is how people coordinate by default nature, and this informal position will move between people as different individuals show initiative. A simple system of ledger books or signing line on various forms will likely be sufficient to keep organized most tasks undertaken informally.

If these positions are not acknowledged, it begins to break down once you reach teams where members are counted in 3 digits, hundreds or several hundreds, simply because now you have to recognize that individuals do not work in groups of such size. So really, you are working with a large group of groups, and the dynamics change quickly. Unfortunately, I cannot comment on how this works in reality. In modern America, by the time a group grows to this size, it is considered a political threat, and ripe for extrajudicial dismemberment by the state. I do not know how it would act in the comparatively safe environment of corporate organization, though I imagine most of the problems could be alleviated by automation and programming, such as a good CMS system, but I have no experience there.