The highway authority of the state, county, city, town, or village may designate and establish controlled-access highways as new and additional facilities or may designate and establish an existing street or highway as included within a controlled-access facility. The state or any of its subdivisions shall have authority to provide for the elimination of intersections at grade of controlled-access facilities with existing state and county roads and city, town, or village streets by grade separation or service road, or by closing off such roads and streets at the right of way boundary line of such controlled-access facility; and after the establishment of any controlled-access facility, no highway or street which is not part of said facility shall intersect the same at grade unless it be an approved interchange. No city, town, or village street, county or state highway, or other public way shall be opened into or connected with any such controlled-access facility without the consent and previous approval of the highway authority in the state, county, city, town, or village having jurisdiction over such controlled-access facility. Such consent and approval shall be given only if the public interest shall be served thereby.