The Government did not invent roads. Roads existed long before the Government made them, in fact most towns and cities had roads without a Government mandating and taxing people for using and building them. If you are referring to the Highway programs, those were not Federal Government ideas. Those were citizen and business owner ideas. The program went to the Feds because it was easy at the time, and saved States from having to negotiate connecting points.
The Government may have expedited some of the process, but we don't know how much because we only implemented one Federal highway program. In other words, it's impossible to measure help or harm from the Federal program. Did it add some benefit, sure, but you can't truthfully claim that it's all because of Government.
There is a period in our history in which most road building was handled by the private sector. The result was a difficult navigate mess of toll roads over much of the Northeast. The interstate highway system was an antidote to that sort of nonsense and has be an incredible success
I'm not sure how many photos you have seen from the 1800s, back before the Government handled trash pickup, but I have never seen any that show giant trash piles in every lot. As with roads, trash pickup was happening without Government intervention as well. The Government didn't come up with concepts like "If you drink water with trash in it, it's not good water", we knew that well before a take over by the Government.
Actually in cities of substantial size garbage removal was a serious problem. New York in the late 1800s like most large cities in world history STANK and was infested by rats. The government may not have invented a desire for clean water but there is no question that it took laws enacted by the government to stop private industry from putting garbage in our water, and air, and everywhere else.
Your last example is the worst. Firefighters used to be all volunteers, and many fire departments still run on a measurable percentage of volunteers. Large cities collect taxes for dedicated people, and people can choose to live there or out in the sticks where they lack the services and don't pay the premiums. Believe it or not, Firefighting has happened in communities for as long as we have had communities without Government intervention.
There is a great example of a big city that had private firefighting. London. If you were able to purchase fire insurance your insurance company provided fire protection. As a result there were as many different firefighting companies as insurance companies and they competed often with disastrous results, while the poor were left to the mercies of whatever ad hoc bucket brigade might form up. It was in response to the tremendous failure of this sort of fire protection that public sector firefighting was started.
In all of your examples, there is not a single case where you can claim that Government is needed. You can in some cases claim it adds benefits, but at the same time it's difficult to measure how much. Road building (construction in general) has, and historically has had, significant levels of political corruption.
It's impossible to provide hundreds of pages of concept in a post, so I'll recommend you read Stephan Molyneux or listen to his podcasts on anarchism. I don't agree with him on everything, but it's good for the brain to contemplate alternative opinion.
I'm sorry but you are just incorrect. In each of the cases that you cite it took government to represent the interested of the whole society to fix the problems created by the private sector approach. There are simply some things that make sense for the people to control.