that is one very strict definition of socialism, which is not a monolith. Nor is there any agreed Purity test of what is or is not socialism

... with that in mind - following your roads example still

• if roads are paid for, described, prescribed and constructed / maintained by private companies as you say - why is the government involved at all? Isn't it more accurate to say the government owns all roads (in the US- due to eminant domain- all land) and contracts private companies to build them

• companies only exist by the express permission of and after registration by the government, and we can argue who holds the most soft power, but the fact is if you fuck up bad enough the government will disband your company for you

• the existence of a market does not mean socialism is not happening: in reality, the "profit incentive" of capitalism is also tempered by the social contract of socialism. In my post I was careful to give examples of the social contract that outway pure profit incentives (ie you can't build a factory in a national park)

• I would say the process of agreeing to do something in exchange for money is neither (/unknowably) capitalist or socialist (or neither or both) without further context.