I did this type of Nuzlock in Crystal Version. I was able to use “gifted” Pokémon and eggs. I started with Totodile and named him Steve. I got Togepi and named him Benedict. I hatched a shiny Pichu from the daycare couple which I simply named Shiny. I kept the Spearow from the guard that was already named Kenya and never delivered the letter. I got an Eevee that I named Fluffy. And I also kept Shuckie the Shuckle. Benedict was lost to Lance and I lost everything except Steve, Shiny, and Shuckie to Red.
36:05 Pokémon Journeys Episode 89-90. This arc alone added quite a few Pokémon to Ash's roster of owned Pokémon, including Pichu, Dragonair, Dratini, Haunter, and Gastly.
Shouldn't you jusf never get wild pokemon since no chance to encounter them. Unless you think that encountering wild pokemon in every single grass is the worst option.