Mensa does administer its own tests. They simply accept LSAT and other standardized test scores in lieu of taking its tests. It’s not laziness on the part of Mensa but rather their accepting that some people who want to join won’t want to bother taking the Mensa test, so they decided to accept other scores instead; they thereby get more people to join than otherwise might.
No standardized test measures IQ anyway. It’s just a benchmark by which the organization - a college, Mensa, law school, etc. - can go by to “measure” everyone on an even scale.