Kinda a hard argument to say the government can do something cheaper than private industry.
Turns out, governments can be efficient and run things well. You just have to tune the incentive structure correctly.
(Aside, I don't care about cheaply as an axis nearly much as I care about "for the benefit of many". I'd take expensive programs that help everyone, every time.)
The interstate highway system, which replaced a complex system of partially privately owned roads, is another.
In most countries, healthcare is another great example. And arguably education as well - public schooling produces generally pretty good results worldwide for not that much money per student, and outside the US, higher education is often public and incredible value for money.
Road maintenance is normally cheaper/faster with private.
Healthcare is a can of worms but you could be right. The real problem with public health is what happens when these gov can't afford it anymore. US is likely in the wrong now but I want to see the results 50yrs from now when social spending cannot keep up with rising costs and decreasing taxes in EU (population collapse).
Broadband. Look at municipal broadband in Chattanooga. EPB is cheap and offers high quality service.
Health care. Typically socialized systems are cheaper. Medicare pays one tenth the administrative costs private health insurance does.
Insurance. National flood insurance for instance simply isn't offered privately because private insurers charge so much.
Loans, in particular student loans. Private loans are so expensive relative to government ones.
Transit. I use the bus and the train when I'm not biking. It costs me $3 to take a 60 minute train to the airport. It costs me $120 to take a 45 minute Uber. Trains are capable of managing stadium traffic, commute traffic, I can put my bike on board. They are clean and safe.
Charging a consumer or taxing them, what's the difference