Don't expect the puller to be in the Briggs and Stratton manual, they did not build the mower, just the basic engine.
The manual that Snapper has basically showed operating and basic maintenance along with exploded view parts diagrams.
As I recall from being in an actual small engine shop back then, many tools were usually fabricated by the repair shops.
We used to remove the 2 set screws and then place a bolt in the center hole so it would contact the tip of the engine crankshaft. From there a 3-jaw puller would be used to apply light pressure. If there was rust, it took patience and lots of soaking to work them off.
Clean the crankshaft end then apply anti seize to the crankshaft end during reassembly.
These do have a height sitting for them to work correctly and work properly. I will see if I can locate my old repair information and/or a Jim Jackson you tube video that I watched not long ago.
Jim Jackson has a hobby/business working on these:
www.bing.com