Is there like an ‘institute of ancient Greek texts’ that publishes our translated Greek books?
There are several of them, though they are private concerns (publishers) rather than government institutes. The Classical Greek canon is not in fact that big (Nick Nicholas’ answer to How much writing from ancient Greece is preserved? Is it a finite amount that someone could potentially read?), and publishers can set themselves the task of publishing the lot in translation, or more typically in bilingual editions. In English, as other answers have noted, the standard-bearer is Loeb from Harvard University Press, which is now available online for subscription. In French, it’s Les Belles Lettres. In Modern Greek, it’s Kaktos.
My former employer the Thesaurus Linguae Graecae did link ancient texts on its site to online translations into English where available; for canonical texts, they almost always were. The database of translations was never under my purview; feel free to bring any misalignments to their attention.