It's not very accurate unless we are talking about the very beginnings of his career.

...there is an upstart crow, beautified with our feathers, that with his tiger's heart wrapped in a player's hide supposes he is as well able to bombast out a blank verse as the best of you, and being an absolute Johannes factotum is in his own conceit the only Shake-scene in a country.

• Robert Greene, A Groatsworth of Wit, 1592 edited for modern spelling PDF

What you just read is the first known mention of William Shakespeare as a playwright. Robert Greene is often called England's first celebrity author and he was not very fond of Shakespeare. In 1592 the man we now call The Bard of Avon would have been writing for less than 3 years with 4-7 plays under his belt.

After 1594 it is safe to presume that Shakespeare was among the most well-known playwrights in the London theatre scene because he had become the in-house playwright for the Lord Chamberlain's Men of which he owned a 12.5% share. The LCM was a big deal as acting troops went. They already had England's most sought-after actor. By 1599 they had England's best theatre. By 1603 they had become King James I's (and therefore England's) official acting company. By 1607, chances were pretty good that if you knew the name of only one English playwright, that name was either William Shakespeare or Ben Johnson.

Besides his official royal endorsements, Shakespeare and the Lord Chamberlain's Men/Kings Men were in high demand outside of the public playhouses where just about anyone could afford a ticket. There are numerous references to private performances in the homes of some of the wealthiest and most prominent families in Britain at the time. The first known performance of A Midsummer Night's Dream was at Court in 1605.

Shakespeare's works were first compiled into a "complete" set with the First Folio of 1623. These books were quite expensive at the time and a second edition was printed a mere 9 years later, indicating that Shakespeare was held in unusually high regard for a playwright. The idea of Shakespeare's plays being "looked down upon" mostly arose in the early 1700s when the overt sexuality and low-brow humor found throughout his works were frequently omitted and many of the tragedies were given happy endings that the author never intended.

The religious middle-class looked down upon the theatre, seeing it as a breeding ground for immorality and social instability, so the professions of actors and playwrights were seen as just one step above vagrancy. In fact, it was the vagrancy laws that dictated that players be sponsored by a member of the nobility responsible to the crown (originally as a way of halting the performance of religious plays, which were a staple of Catholicism, and hence by law traitorous). Poets, however, were looked upon much more favorably. While Shakespeare's early poetry was praised, his career as an actor and playwright was more often than not looked upon as unfortunate. A couple of poems by John Davies illustrates this dichotomy:

To Our English Terence, Mr Will. Shake-speare

Some say (good Will), which I, in sport, do sing,

Hadst thou not played some Kingly parts in sport,

Thou hadst been a companion for a King;

And been a King among the meaner sort.

Some others rail; but, rail as they think fit,

Thou hast no railing, but, a reigning Wit:

And honesty thou sowst, which they do reap;

So, to increase their stock which they do keep.

IOW, Davies reports that some persons said that if “good Will” had not been an actor, he could have been a companion to a King. (This goes back to Shakespeare's putative ambition to be dubbed a knight by King James, who was selling knighthoods to raise money.)

Another is from Davies' Microcosmos (1603):

Players, I love ye, and your quality,

As ye are men, that pass time not abused.

And some I love for painting, poesy (W.S. R.B.)

And say fell fortune cannot be excused,

That hath for better uses you refused:

Wit, courage, good shape, good parts and all good,

As long as all these goods are no worse used.

And though the stage doth stain pure gentle blood,

Yet generous ye are in mind and mood.

W.S. - William Shakespeare and R.B. = Richard Burbage, who was a painter as well as an actor.

So being a good playwright was a bit like being the most beautiful whore in town: you got lots of attention from the high and the rich, but they certainly weren't going to let you marry into the family. Sir Richard Baker probably sums it up best in his Chronicle of the Kings of England (1643):

"... it might be thought ridiculous to speak of stage players; but seeing excellency in the meanest things deserves remembering ... For writers of plays, and such as had been players themselves, William Shakespeare and Benjamin Jonson have specially left their names recommended to posterity."