That person is called a construction supervisor: someone hired by the owner who makes sure the contractor is doing good work and not ripping you off, as well as keeping the owner in line and making sure they don’t get in the contractor’s way, delay decisions etc.
Around here they aren’t interested in projects under $5M.
In some parts of the world, mine in Europe included, those guys are required by law and are usually employed by the municipality (but the construction project owners are charged for their services via a special fee).
They take on some personal responsibility for the way the construction project is handled (at least for the things that fall within their purview).
That sounds like building inspectors in the US. They vary widely in competence, but all will ensure the permit fees are paid and property taxes raised; most will ensure the work is done to code; approximately none will ensure the work is built to plan details.
That is annoying part - even if you could overspend and match salary for the guy - it still not guaranteed they will work for you. Mostly because anyone who is good is swamped with requests and can choose what they will work on.
Around here they aren’t interested in projects under $5M.