Consequently there are a lot of gaps in this approach. For example, wahapedia is a mirror for Warhammer 40k rules. Games Workshop, the rule book authors, would be happy for the site to be inaccessible, but since the site is hosted in Russia, and they don’t respect western IP laws, it continues to stay up.
The less the government has to do, the lower your taxes in theory. Exporting government functions to private institutions is very common. For example, many legal agreements have forced arbitration clauses. It is perfectly acceptable for two consenting parties to agree to alternative dispute resolutions outside of court. Except one of those parties (the customer) has no ability to influence the wording of the contract and all vendors have the same arbitration clauses. Consumers pay for it in the form of lower value and higher prices.
Many municipalities require new construction to have homeowners associations. They don’t want to have to deal with all the resident disagreements. These HOAs charge a fee to residents for the upkeep of the services in the community (like a government), and have legal means to enforce transgressions such as foreclosure on your home (like a government). But there is also another government over it, they still charge you property taxes (which ostensibly pays for those services) and those taxes haven’t gone down.
So yes, not only is moving government functions to private entities inefficient, and there’s plenty of examples of to back it up, it just shifts responsibility around and ultimately costs regular people more.