Is this actually the case? Philosophers seem to consider Aristotle really influential and quite outdated, while English lit. people who aren't decrying him as a dead white male patriarchal oppressor still seem to think Shakespeare was probably the greatest thing ever.

If you asked some modern philosophers to make a list of the greatest philosophers of all time, and then asked English lit professors to make a list of the greatest English writers of all time, the relative rankings of Aristotle and Shakespeare would likely be close to each other (though probably not identical). And I think the reason for this is that they're ranking -- and teaching, and recommending -- mainly along lines of influence rather than technical skill or correctness or enjoyment. The precise terms each are described in might be different, but in terms of their place in their fields I think the analogy's pretty close.

I do think that Shakespeare by most standards would look better than Aristotle relative to his counterparts today: literature was a more mature field in his time than philosophy was in Aristotle's. But I don't think he was the most technically skilled writer in English, not by a long shot, and I suspect most literary scholars (Shakespeare scholars excepted) would agree with me.