It's true that Objectivism "starts" with the primacy of existence, as contrasted with the primacy of consciousness. Your "conscious experience of determinate objects" presupposes the primacy of existence. If your "experience" is not experience of existence ("determinate objects"), then what is it experience of -- non-existence? If "experience" is not a consciousness, then is it a non-consciousness? Consciousness prior to existence is consciousness of non-existence. But prior to or without existence, there is no consciousness -- what is it conscious of, if not existence?. Consciousness conscious of nothing but itself, without prior consciousness of existence, is consciousness of nothing, is no consciousness.