The excessive movement in the nose gear as shown in the video posted by the OP is indicative of metal deformation at the location where the nose gear is bolted to the engine mount. There are 3 steel items at that location that could potentially yield leading to the movement. The first is the spring steel nose leg, the second is the engine mount, and the third is the AN bolt. I am sceptical that replacing the bolt would produce a satisfactory outcome because I would be reasonably confident that the yield strength of the bolt would be considerably higher than the yield strength of either the nose leg or the engine mount. My best guess is that in this case there would be yielding/deformation at the 2 holes in the engine mount.