Would it be unbalanced for Dex-based Fighters to choose proficiency in Dex saving throws instead of Str saving throws?

Whilst building a Dex-based Fighter (an Arcane Archer), I decided to pick the Resilient feat at level 4 so that I could have proficiency in Dexterity saving throws, which makes sense given that they are a Dex-based character. Then I thought about how odd it was that they had proficiency in Strength saving throws just because they’re a Fighter even though they’re not a Str-based character. Sure, it makes sense for a lot of Fighters, but not all of them.

Therefore, I’m considering introducing a new homebrew rule for whenever I’m running a game and a player of mine wants to make a Dex-based Fighter:

Saving Throws: Strength or Dexterity (your choice), Constitution

The “choice” would be made at level 1 (I don’t plan on allowing them to switch it back and forth).

Given that this class is the only one listed under the Multiclassing section in the PHB (pg. 163) has having an “or” in their requirements (“Strength 13 or Dexterity 13″), this seems to fit the intent that Fighters aren’t tied to Strength.

The Battlemaster archetype (PHB, pg. 73) also allows either for the saving throw for some maneuvers, again implying that Fighters are supposed to be flexible regarding using Strength or Dexterity:

Maneuver save DC: 8 + your proficiency bonus + your Strength or Dexterity modifier (your choice)”

Would there be any problems with this? They would still be outclassed in Dexterity saving throws by Rogues and Monks once they get Evasion, so I don’t see this stepping on their toes too much (at least not at higher levels), but on the other hand no class RAW allows a choice in saving throw proficiency like this, so would there be any other problems I’m overlooking?