Using egg yolk is similar to brain tanning. You're coating the protein fibers with fat so they slide past one another. When the individual parts slide smoothly, the hide feels soft. Problem with egg/brain tanning is that it's not very waterproof. Most traditional recipes include some hot smoking so you use a fire's heat to set the proteins in the skin and the smoke adds some color and antibacterial properties. Also the fats in the egg may go rancid and the protein that gets smeared on (part of the yolk is protein) may be sticky if it gets at all wet.

The recipe above uses acid and salt to cross-link and set the proteins in the hide instead of heat. And the neats-foot oil is like the fats of brain or egg yolk.

Using acid to tan a hide is fairly safe. Compared to chromium salts and other types of tanning solutions, which you can't dispose of easily, a salt-acid solution can be dealt with in most urban and rural situations. Acid can be diluted an poured down the drain. I wouldn't do that on a septic system.... You can also try drying it in a pan to re-concentrate it if you're losing ionic strength.

My suggestion is to pour the acid slowly and wear safety glasses. Also, litmus paper would make it feel less like a backwoods farm project and more like chemistry (it's a matter of style).