Aluminium antiperspirant stains are actually pretty easy to remove if you soak your shirts in something acidic - I use sulphamic acid (commonly sold as the main ingredient in coffee machine descaler, so pretty safe, and not as smelly as vinegar) and just leave things for a day, then any residue that’s left just comes out in the wash. The reason it stains is that long-acting antiperspirant seals the aluminium salts in tiny wax beads which break open over the course of the day to keep releasing more, and wax needs a lot of exposure to soapy water to dissolve, but it’s protected by the aluminium salts jellifying when wet, and they in turn are protected from rinsing away by the surrounding wax. Acids can attack both.