For me, the oven simply works better than the stove. It is the stove that you can skip.
First, the heat. It is the same principle as with many foods. A pan heats quicker, but with more variability. An oven is slow and steady, and reaches equilibrium more easily - and it is much easier to control the equilibrium temperature. On a stove, especially empty pan, a low heat will not be enough to polymerize the oil, so it is tempting to turn up to high heat - but then you have to manage to not overheat it and either burn away the seasoning while it is happening, or produce a different reaction - I don't know which one it is, but I suspect it may be the non-rust iron oxide, from the looks of it.
Second, the layer thickness. If your oil layer is too thick, you don't get seasoning, you get a puddle of hot oil. If you continue heating that for too long, you get a thick crust of soft polymer that is prone to peeling off, instead of a thin layer that is well-stuck. A thin layer is optimal, but difficult to produce - the usual method of letting the oil spread by flowing (aided by tilting the pan) makes it too thick. Smearing it with a towel (paper or cloth) gives you a good thickness, but you get a bit of lint onto the pan, especially if the surface is rough, like cast iron. What works well is to cover the pan well with the tilting method, and then have the excess drip slowly when heated to low enough viscosity - which happens automatically during the upside-down seasoning in the oven.
Third, there is the "how to season the sides" problem that Sneftel covered.
Fourth, if you don't have a hob that is the exact same size as the pan, the high temperatures on the empty pan can easily cause warping, especially in thinner forged iron pans.
Fifth, seasoning on a stove is a very active process. You have to manage the heat, keep an eye out for an overheating (warping or fire risk), look for hot spots and possibly rotate the pan, and look out for oil pooling into an undesirable thick layer. In the oven, you set it and forget it.
In summary, the oven is simply the much friendlier way to do it.