MMA welding of leaded sulphurised and phosphorised steels is possible but not easy. MIG and TIG is a non starter. The problem is that lead sulphur and phosphorus in free machining steels, are low melting point alloys. During the welding process, the steel begins to solidify from the outside of the weld inwards, this causes the lower melting point alloys to migrate to the centre of the weld. If they are present in sufficient quantity, weld porosity occurs as they solidify and shrink and lead to what is termed centreline cracking. This will not be visible but NDT will show microporous cracks within the weld line causing a weak weld. Therefore free machining steels are deemed unweldable as they are not structurally sound.
There are ways to overcome this but there are no guarantees of the weld integrity. Using a stainless rod helps with the weld process as it helps to prevent cracking, there are also specialist carbon steel rods available.
TBH I would avoid welding free machining steels unless absolutely necessary and certainly would not use them in stressed or structural applications.