The quote in the Question looks like a highly approximate summary of Berkeley's idealism - esse est percipi : to be [exist] is to be perceived - but Berkeley did not link epistemology and ontology so tightly as the Question quote suggests.
The 'esse est percipi' principle first appears in Berkeley's Philosophical Commentaries (Entry 429). We should note that in The Principles of Human Knowledge it makes its initial appearance in I §2 : 'the existence of an idea consists in being perceived'. It is only by a course of argument that Berkeley infers that sensible objects are nothing but (what he early on calls) 'collections of ideas' (Principles, Intro.§1) and hence that for them too, as ideas, to be is to be perceived.
A fuller version of the principle is 'esse est percipi aut percipere' : to be is to be perceived or to perceive (Philosophical Commentaries, Entry 429). But the shorter version is right for the Question.
▻ TO BE IS TO BE PERCEIVED DOES NOT MEAN OR ENTAIL THAT TO BE IS TO BE LOOKED AT
'To be is to be perceived' means just that : perception is not confined to sight and looking. We can perceive by or through any of the senses. I can perceive a table by touching it; I can perceive a door, when I am blindfolded, by its resistance.
▻ TO BE IS TO BE PERCEIVED DOES NOT MEAN OR ENTAIL THAT TO BE IS TO BE PERCEIVED BY US
Sensible things, objects of perception, Berkeley tells us :
... cannot exist otherwise than in a mind perceiving them. I think an intuitive knowledge may be obtained of this, by any one that shall attend to what is meant by the term exist when applied to sensible things. The table I write on, I say, exists, that is, I see and feel it; and if I were out of my study I should say it existed, meaning thereby . . . that some other spirit actually does perceive it. (The Principles of Human Knowledge, I §3.)
That 'other spirit' could be non-human and a likely candidate is God :
'Since sensible things do not have independent existence in Berkeley's view but exist only as collections or congeries of ideas in the mind, he concludes that some sensible things must exist continuously in the infinite mind of God, at least during those intervals of time when they are not perceived by finite minds' (Dale Jacquette, 'Berkeley's Continuity Argument for the Existence of God', The Journal of Religion, Vol. 65, No. 1 (Jan., 1985), 1.)
So ontology (the existence of sensible objects) is not tied to epistemology (human perception). Hence my remark at tat start that Berkeley did not link epistemology and ontology so tightly as the Question quote suggests. But we might note that 'some other spirit' does not logically imply that it is the same other spirit, namely God. There could be a multiplicity of such spirits.
▻ HISTORICAL ASIDE - BERKELEY AND THE GREEKS : PLATO
The Question mentions a possible anticipation of Berkeley's view in Ancient Greek philosophy. Plato's Theaetetus comes to mind. Here's Miles Burnyeat on the point :
In the first part of the dialogue a theory is elaborated according to which nothing exists outside the particular perceptual encounter in which it appears to sense. (M. F. Burnyeat, 'Idealism and Greek Philosophy: What Descartes Saw and Berkeley Missed', The Philosophical Review, Vol. 91, No. 1 (Jan., 1982), 4.)
The context is Theaetetus' definition (151e) of knowledge as perception.
▻ TWO PROBLEMS - (1)
Berkeley takes for granted that sensible things exist continuously when not perceived by finite minds. There is no logical necessity to assume this. Perhaps they do pass into and out of existence according as they are perceived by finite minds. It may be counter-intuitive, but so ?
▻ TWO PROBLEMS - (2)
We are all familiar with the 'myth of the given', we know about the theory-ladnenness of observation, we know that perception involves inference (as when I see that a poker is hot). But there is, or so it seems to me, an irreducibly passive element in perception. If we are to perceive something then, it strikes me, there must be something to be perceived, something which is not purely a product of our own minds. Something is given in perception - presented to consciousness - if nothing like as much as was once widely thought.
Now, this does not fit well with the idea of God, 'that active principle' (Principles, I §66). God is on Berkeley's account a purely and wholly active mind or spirit. Nothing is or can be presented to the consciousness of such a being. Therefore God does not perceive and is incapable of perception. It follows that God does not perceive sensible objects, and ensure their continuous existence, in our absence.
G. Berkeley, The Principles of Human Knowledge, 1710.
G. Dicker, Berkeley's Idealism: A Critical Examination
ISBN 10: 0195381459 / ISBN 13: 9780195381450
Published by Oxford University Press, 2011
Dale Jacquette, 'Berkeley's Continuity Argument for the Existence of God', The Journal of Religion, Vol. 65, No. 1 (Jan., 1985).
Jonathan Bennett, 'Berkeley and God', Philosophy, Vol. 40, No. 153 (Jul., 1965), pp. 207-221.
George Pitcher, Berkeley, London : Routledge, 1977.