The only thing that raises the odds of finding a shiny legendary is the shiny charm, which improves your chance from 1 in 4,092 to 1 in 1,365.

What does the shiny charm affect?

The Shiny Charm increases the chance of encountering a Shiny Pokémon, both in the wild and through breeding.

Is it worth shiny hunting without the shiny charm?

While collecting the shiny charm is not explicitly required to find a Shiny Pokémon, it will help you out immensely, as it essentially doubles your chances of finding Shiny Pokémon across the board.