Here's what I noticed with a little analysis. How much pressure do we want to put on players to chop Forests early on?
Right now, if you have a Grassland/Forest with access to fresh water, the only pre-Medieval way to get it to food-neutrality is a Farm. (Once you get to the Medieval Era, you have more choices; Watermill is food-neutral but a Farm with Crop Rotation is food-positive.)
If you give a Forest +1 food with an improvement that doesn't destroy the Forest, that also makes it food-neutral, relieving the need to chop the Forest to fully support 1 citizen. +1 food/-1 production (total yield +3 food) is worse than a riverside Farm (+3 food/+1 commerce) but equal to a lakeside irrigated Farm. I think that would be fair, if we coupled it with tech bonuses that make it not as food-focused as a Farm in the long run. So the improvement wouldn't see a food bonus for a long time. The second Farm food bonus comes at Organic Chemistry, so that's around where the forest-harvester would get a food bonus. On the other hand, what is the real use of a tile that does nothing other than support a citizen? A population point tied up in a tile is usable for drafting or growing a city up to its cap, but I think that's about it.
The two things I want to avoid are marginalizing Farms and increasing Farms any further. Farms are the center point around which all the other improvements are balanced.