I have a 2m x 2m floor tile between my hearth and my bed on which a branch spawns every couple of days. Free wood!

I don't use any mods and your situation is an example of why. The modder figured out how to change the properties of mushrooms and thistle to enable them to be planted. The modder did not examine the algorithm for carrots or turnips to determine what prevents the planted versions from spontaneously respawning.

Like many a quick fix, undesirable consequences have arisen. So you replaced one problem (scarcity of a resource) with another (overabundance of a resource). As the saying goes, you pays your money and you takes your chances.