That statement is a truism of course, therefore agreed, although may not be related to success. I have an adage I use personally, "It only takes one to have an audience". If occurs, then IMO I am successful 'as' a writer conveying a message. One may play with that however they wish.
Once measures are introduced then the definition of success for many changes. Possibly it evolves. I can run and am successful until I do not run fast or far enough. But, still, I can successfully run.
For me after a time of study I realized there is a difference between being a writer and an author. Even though used many times synonymously, IMO they are not. Not meaning to go off tangent let's take a peek. Using Wikipedia we discover:
"An author is 'broadly' defined as "the person who originated or 'gave existence to anything' and whose authorship determines responsibility for what was created. 'Narrowly' defined, an author is the originator of any written work and can also be described as a writer."
I prefer the broad stroke of the brush . . . the big picture view with success. I prefer 'anything' itself may mean 'my adventure as a writer' . . . after all I created that. So, we may discover a painter, a sculptor, a writer, or even a cabinet maker 'can' be an author of a work. So, what are the keys? For me one is the meaning of -ship:
"a native English suffix of nouns denoting condition, character, office, skill, etc"
When I play with that I think of my 'ship' that I sail on the oceans and seas of writing. When the currents are calm and the winds in the sails are full I head toward the Island of Success easily and quickly. But, like most destinations upon the great oceans and seas that destination is never seen while sailing. It is beyond the horizon.
I must navigate the currents both turbulent and calm. I must understand the weather. Mr. Sunshine will open his wide arms of warmth stretching across the brazen blue horizon. Or, it may be a torrential down pour with looming clouds of obscurity. There may be sudden squalls pushing me off course. I, still, forge forward navigating those changes seeking a point on the horizon. And, if not arrived upon I discover both comfort and confidence I know the Island Success is where it always will be . . . beyond the horizon.