The one advantage of Autotransformer starting is that the transformer amplifies the motor current when dropping the voltage so you do end up with more starting torque PU of line current. That has to be weighed against the higher cost, larger physical size, higher weight, limited starting time, inflexibility and tendency to wear out when compared with solid state.
I also used to say that any electro-mechanical starting method has the advantage of unsophisticated users being able to watch things happen, which gives some people comfort I guess. But with current Arc Flash regulations, that's no longer possible anyway.
Now if, by saying "magnetic soft starters" you were referring to the one single type still sold in India after being abandon by the rest of the world 50 years ago, but trumpeted as "new" again, I will not further engage. To do so would require I use the term they coined for it, then anyone searching for it will end up being directed to their website because they are the sole manufacturer. That company has used this forum, and others, as a form of free advertising by posting "questions" and eventually dropping their trade name, indirectly driving hits to their site. If they want advertising, they should pay for it like everyone else.
"You measure the size of the accomplishment by the obstacles you had to overcome to reach your goals" -- Booker T. Washington