Are there public functions that might benefit from more competition, including competition from private firms?

Yes there are. One example would be Government involvement in taking care of illegal immigration.

Illegal immigrants are not entitled for the minimum wages that are set by the Government. So, the companies often agressively seeking for them so they can pay them cheaply for low-level tasks, and give the company a decrease in total expense. (which will give a competitive advantages toward companies that use legal workkers)