Partly correct, but it isn't automatically snobbery that is what leads this. Snobbery about aristocratic trappings is used as (very flawed) evidence against Shakespeare.
No, what it is is an attempt to fit Shakespeare into what had, relatively recently, defined a poet.
You see, in Shakespeare's era, he was considered a Playwright. That's a fanciful title if one were to call oneself that today if you wrote "Starlight Express", but in the day it was merely a mundane job title: One who manufactures plays, not terribly different than a Cartwright or Wheelwright.
And that's pretty much how all poets and authors tended to be in that day. Craftsmen not looked upon as any different as any other.
Then came Lord Byron.
Lord Byron and his works ended up redefining what a poet was. No longer was it merely some guy penning it up on the 2nd floor while his grain sales down below are putting food on the table. Now your poet had to be independently wealthy, but more important he had to be noble, flawed and tortured.
So here's this son of a glovemaker producing some of the most referenced works in the English language, and in the late 19th century (and early 20th century) and he just does not fit with their Byronic Ideal.
So another candidate must be found. And we wind our merry way down the list of candidates. Francis Bacon was always the favorite until recently, since he wasn't inconveniently dead when a large number of the plays were written. But in recent years Oxfordians have come out of the woodwork thanks to a lousy movie that thinks Marlow couldn't write in Iambic Pentameter.
If you need proof, look no further than the OP where he refers to Oxford as "a flawed and even tortured genius", filling that Byronic checklist nicely once you note the Earldom at the head of his name.