It does also apply to Natural 1s, because of the exact text and specific overrides general.
The Brute Martial Archetype's 7th level feature, Brutish Durability, reads:
Whenever you make a saving throw, roll 1d6 and add the die to your saving throw total. If applying this bonus to a death saving throw increases the total to 20 or higher, you gain the benefits of rolling a 20 on the d20.
The wording here leads me to believe that this is the specific that overrides the general rule of rolling a 1 on a death saving throw.
However, it's not as good as you might think with a natural 1. Adding a d6 to the roll will rarely ever increase the total to 20 or higher, the only case I can think of (Barring Monk's 14th level Diamond Soul because this feature is 7th level) is being attuned to a Cloak of Protection for +1 to saving throws, Ring of Protection for +1 to saving throws, and Stone of Good Luck for +1 to saving throws and rolling a 6 on the Brutish Durability die while being in the Aura of Protection from a Paladin with 30 Charisma, for a grand total of exactly 20.
Because the wording of the feature only has an affect if the total reaches 20, it wouldn't automatically prevent you from taking two failures a natural 1 would normally bring.