It would amount to assault or some other type of forcible intervention. While the people who work or seek the services of the facility's operators view themselves as having the right to make use of the public road or street that leads to the facility, the demonstrators also believe that they have this right. When their purposes clash, there is inevitable conflict. For one, if public authorities kept the purpose for which a public sphere exists clear and unambiguous, this might help. Streets, roads, sidewalks, parks and such could all be kept as areas where the kind of activities proper to their design would be conducted.
Streets are for transportation, except in some special cases as when a parade is being held.
Roads are for vehicle travel, sidewalks for pedestrian travel and parks for recreation. If someone were to set up a printing press in the middle of New York City's Fifth Avenue, this could be stopped and no one's rights would be violated. If some people were to try to hold a picnic in the middle of a sidewalk and the police removed them, this again would not be a violation of their rights. Only because of the misconception that public spheres may be used for anything by any member of the public. In fact, the prohibition against regulating free speech by Congress is not a prohibition against keeping public facilities to be used for their designated purpose.