Tragedy of the commons is more applicable to capitalism, it's competing groups trying to get as much resourses as possible when there is limited resources.

Public works can be done on a cooperative basis, by unions of workers.
The Conquest of Bread is a book that outlines how an anarchic society may function, here's an excerpt from it about rails:

In support of our view we have already mentioned railways, and we will now return to them.

We know that Europe has a system of railways, over 175,000 miles long, and that on this network you can nowadays travel from north to south, from east to west, from Madrid to Petersburg, and from Calais to Constantinople, without delays, without even changing carriages (when you travel by express). More than that: a parcel deposited at a station will find its addressee anywhere, in Turkey or in Central Asia, without more formality needed for sending it than writing its destination on a bit of paper.

This result might have been obtained in two ways. A Napoleon, a Bismarck, or some potentate having conquered Europe, would from Paris, Berlin, or Rome, draw a railway map and regulate the hours of the trains. The Russian Tsar Nicholas I. dreamt of such a power. When he was shown rough drafts of railways between Moscow and Petersburg, he seized a ruler and drew on the map of Russia a straight line between these two capitals, saying, “Here is the plan.” And the road was built in a straight line, filling in deep ravines, building bridges of a giddy height, which had to be abandoned a few years later, after the railway had cost about 120,000 to 150,000 pounds per English mile.

This is one way, but happily things were managed differently. Railways were constructed piece by piece, the pieces were joined together, and the hundred different companies, to whom these pieces belonged, gradually came to an understanding concerning the arrival and departure of their trains, and the running of carriages on their rails, from all countries, without unloading merchandise as it passes from one network to another.

All this was done by free agreement, by exchange of letters and proposals, and by congresses at which delegates met to discuss well specified special points, and to come to an agreement about them, but not to make laws. After the congress was over, the delegates returned to their respective companies, not with a law, but with the draft of a contract to be accepted or rejected.

If capitalist rail companies can cooperate to build a rail system, rail companies owned by the workers would cooperate much more freely.

Anarchic societies actually saw production increase, since it eliminated a lot of useless jobs, In an anarchist reigon of spain, they produced so much bread and oil that after giving it away for free they were still able to export some (source). (I highly recommend you read Bullshit Jobs by David Graeber, its a great book).

If people were able to overthrow a government once, they can surely do it again for a warlord. If anything, it would be harder to re-establish a government since people will see their lives materially improve with anarchism. Outside forces are a problem, but they're a problem with capitalism as well.