Recently heard in a video about Babylonia that a date plantation was up to 4x more productive in calories than a comparable wheat plantation, tho took 7 years to mature. Were there any societies whose staple food was derived from silvoculture? Or more broadly, not from grasses or tubers?

I don't think it's quite right to think that Babylonia ran on dates instead of barley - they were both important sources of food and recognised as such in surviving texts. There's an overview in Adams 1978 (slightly outdated now but convenient).

The importance of cultivated grains is abundantly clear from the Code of Hammurabi, which extensively discusses the rules surrounding inheritance and sale of grain fields (36-50), as well as neglect thereof (55-59). There's also direct acknowledgement of the time it takes to plant an orchard, with related rulings:

(59) If a man cut down a tree in a man's orchard, without the consent of the owner of the orchard, he shall pay one-half mina of silver. (60) If a man give a field to a gardener to plant as an orchard and the gardener plant the orchard and care for the orchard four years, in the fifth year the owner of the orchard and the gardener shall share equally; the owner of the orchard shall mark off his portion and take it.

So the value of a single fruit tree was half a mina (30 shekels - triple the fine for causing a miscarriage), and the text implies that it takes four years for an orchard to become profitable.

Outside of Babylonia (but still in the ancient Near East), the Hebrew Bible acknowledges the important of fruit trees in one odd rule, on dealing with captured cities, in Deuteronomy 20:19-20:

(19) If you besiege a town for a long time, making war against it in order to take it, you must not destroy its trees by wielding an axe against them. Although you may take food from them, you must not cut them down. Are trees in the field human beings that they should come under siege from you? (20) You may destroy only the trees that you know do not produce food; you may cut them down for use in building siege-works against the town that makes war with you, until it falls.

The context of this is that the text discusses capturing cities that were considered part of the Israelites' promised land, so after eradicating all the inhabitants of the city (v. 16), the city is to be resettled by Israelites, who of course need to have something to eat - hence keeping the fruit trees, which otherwise would take years to grow back.