In addition, my understanding is that while Shakespeare was well regarded during his life, he wasn't particularly held in much higher regard than many of his contemporaries. The excessive elevation of Shakespeare (Bardolatry) seems to have been a process that occurred over decades and centuries.

It's gotten to the point where it's hard to have a unbiased opinion. People aren't given a copy of King Lear and Tamburlaine and asked which one they prefer. People are usually told ahead of time that Shakespeare plays are wonderful works, that viewing them as wonderful is a sign of intelligence, and people are usually exposed to many plays by Shakespeare and none by his contemporaries.

Our society probably has a much larger impact on how we view these works than the works themselves.

Some big literature classics in Brazil (I suspect other places too, Dickens maybe?) were published as weekly columns on newspapers. Kind of the soap opera of the times.

Now they are treated as high brow art, with the good and bad parts of this stereotype (e.g. sophisticated/boring).