That model works if company A and company B have to pay in its entirety the cost of the building project they have contracted to do, that is to say, if they're working for free.
But yes, Company A that won the contract, and Company B that provided the raw materials, have both added value as a result of building the bridge. Further, the employees of companies A and B will be spending money in companies C,D,E,F and so on, thus adding to their value as well.
It really is no different than Company G, that makes widgets, adding to the value of the raw materials it buys from Company H and making a product, whatever that is.
And, the market for that product is much improved due to the employees of Companies A and B, as well as their stockholders and creditors.
and all of those companies, A-H, being more profitable, will pay more in salaries, in dividends, and in taxes than they would had the bridge never been built.
Private sector highways and bridges? Sorry, that just is not practical.. Oh, darn, there's that word you hate again.
To efficiently build a highway, the government has to exercise imminent domain. Further, the government does not have to post a toll booth at every intersection in order to collect money for its construction and maintenance. Just how else could a highway or bridge project possibly be profitable for private enterprise to build?