a soft reset is resetting your game without deleting anything. a power down or closing the game will count as a soft-reset method; it'll just take longer. Powering down or closing the game works just as well as the button mapped reset.
If it's too annoying, you could always get your L button repaired; it shouldn't be too hard or expensive.
> I noticed the Pokémon’s ID does not change when I do this.
What do you mean by this? the Trainer ID won't change since you're the trainer; it'll be the same for all pokemon you get. That's just to say you're the trainer and generally has no indication if it can be shiny or not.