I’m afraid I have to demur – 18th and 19th-century reception history is not my field, but I don’t know of any time period when Shakespeare’s plays weren’t popular, in the sense of being widely read, performed, referenced, etc. There are definitely eras, including his own, when they weren’t high culture or regarded with particular reverence, but that’s a different thing. Samuel Pepys, for instance, writes about going to see a Shakespeare play in much the same way a modern diarist might write about going to see a movie, and he expresses some critical opinions that, to modern ears, sound entertainingly wrong-headed – but the point is that he is seeing Shakespeare, a lot of Shakespeare, because if you were an avid theatergoer in Restoration England you couldn’t avoid it, and that tells us other people were enjoying Romeo and Juliet and A Midsummer Night’s Dream much more than Pepys did. (He did like Macbeth, and saw it half a dozen times over the course of a decade.) Nobody in Restoration London was going to see a Shakespeare play because it was “good for you.” Theater was pure entertainment, and these plays wouldn’t have been revived if people didn’t find them enjoyable.
What comes later, in the 18th and 19th centuries, is the idea of Shakespeare as awesome-transcendent-genius, pinnacle of English literature, object of reverence, study, and analysis, etc. This is also when people stop adapting plays to suit audiences’ shifting tastes and start caring a lot about the “canonical” version. The version of Macbeth Pepys saw was basically a musical – it amped up the witch scenes and added plenty of singing, dancing and spectacle, because that was what audiences loved. The version of King Lear that dominated the stage until the nineteenth century had a happy-ish ending where Edgar and Cordelia marry (never mind that she’s married already!) although Shakespeare’s original version was always available in print. But that doesn’t mean Shakespeare wasn’t popular, any more than the existence of My Fair Lady means Shaw was unpopular by the 1950s. Playwrights who aren’t popular don’t get their work performed and adapted at all.
Mmm. Depends on the scholar, and on the comic character, but per this website, eight of the twelve most commonly performed Shakespeare plays are comedies – including some, like Merry Wives and The Comedy of Errors, that aren’t particularly regarded as “great” in the literary sense. I don’t think that would be happening if audiences didn’t find them funny on the stage. (And I would disagree that the language poses that much of a barrier, in general – I mean, if you’re watching Macbeth you can see the captain is wounded and covered in blood, and you don’t need a scholar to tell you that bloody means, well, bloody.)
Like I said, I don’t know if this means alt-universe Shakespeare would go over well with audiences or not. (They might be primed to like completely different things; maybe John Milton is THE canonical English author, and, as a consequence, people are attending recitals of epic poetry instead of going to the theater at all!) But, in general, I’d say his popularity in this universe rests on the fact that he knew how to write a cracking good play, and audiences still genuinely enjoy his work.