Kasparov seems to be a respected public intellectual or at least it’s debatable which is more than you can say for most others (though maybe that’s the exception that proves the rule).
Because I find that there's a very wide range among 'well-regarded (by some) public intellectuals.' Some of them say things worth thinking about. Many others, not so much, the only noteworthy thing about them is that they stand on a soapbox.