Welcome to the real world. Besides, how many poor people are flying multiple times per year anyway? I don't have the numbers, but I would bet that this tax would be paid overwhelmingly by businesses on behalf of their employees who travel for work.
And that tax would be absorbed unnoticed in the businesses' expense again right next to the reciepts for over-priced steak dinners and strip clubs. The whole reason airlines can get away with charging so much for business class is because businesses don't generally give a shit if their sales personal are charging them for in-flight champagne when they go to close a billion dollar contract. The truth is that you'll have to be more strategic than randomly throwing a tax at the problem if you want to change the behavior of entities who already dump hundreds of thousands on maintaing a private jet fleet so that a C-level doesn't have to wait in line at security.