Could I issue clearances at a private airstrip without being a controller?
NO!

An uncontrolled field is just that - uncontrolled. No pilot will accept a clearance in uncontrolled airspace in the first place. Consider the legal consequences if two aircraft were to collide as a result of a "clearance" issued by you. Since the aircraft are in uncontrolled airspace, it is the responsibility of the pilots to make sure they do not get too close. However,  if you got on the radio and started issuing "clearances", you might trick pilots into thinking that you are in control. If something bad happens, you should not expect to just walk away from it.

Besides, what makes you think you are qualified to provide air traffic control? Becoming an air traffic controller takes years of intense training, and only the top few percent of applicants are admitted to the education in the first place. Even the fact that you have to ask this question makes it seem like you have a very poor understanding of air traffic control.

The issue of not wanting people to land on a private airstrip has nothing to do with air traffic control. Air traffic controllers do not exist to tell pilots what they can and can't do - they are there to provide a service to the pilots! As others have pointed out, this would be a matter of trespassing. Pilots are very well aware that private airfields are just that - private. No one will just randomly spot a private airfield from the air and decide to land there without having first done some research - unless in emergency situations, of course.