I think 9th level Indomitable is an example of weak design, for two reasons. First is comparing to monk's 14th level Diamond Soul. And second is comparing to non-proficient save DCs around 9th level (when Indomitable becomes available).
Your proposed tweak to make it recharge on a short rest seems reasonable to me. It highlights that the fighter can do this reliably, and "reliability" is the main shtick of the fighter class design.
Diamond Soul comparison
Indomitable is outshined by the monk’s Diamond Soul in every way – except that fighters get Indomitable earlier than monks.
With Indomitable, a fighter can reroll a saving throw 1 to 3 times per day.
With Diamond Soul, a monk can reroll a saving throw multiple times per day and
also gets proficiency in
all saving throws. At 14th level, a monk has 14 ki points, so they could conceivably do this way more than the fighter, but a realistic ballpark is expending ki points three times per day on Diamond Soul.
In this regard, monk and fighter are competing for the same design niche, and the monk is outperforming the fighter when viewed across the entire level span.
Saving Throw DC for Non-Proficient Saves
By the time a fighter gets Indomitable at 9th level, save DCs are 16+, and you have two types of saving throws: Those you have about +9 or better on, and those you have like +2 or worse on.
So you almost always use Indomitable on saves you've only got a 25% chance of success on anyways.
For me, it rarely feels
good to use this ability. Instead it feels like, "Reroll a save you still can't possibly make. Pray for a natural 20!" or "Waste an ability rolling d20+12 vs DC 17. Nice rolling, chump. I bet you won't need this later, right?"
Fixes/Tweaks (for those who thing it could use them)
Making Indomitable recharge on short rests would be one approach.
The original playtest design for Indomitable granted advantage on all saves, IIRC. That would be another approach.
Some people have replaced Indomitable with Legendary Resistances, so instead of a reroll it's an auto-success on a save. Pretty powerful, but that's another approach.
In my latest rewrite of the fighter class, I opted to make it about damage mitigation instead of saving throw boosting – which I'm not 100% on, as it wouldn't apply to charm effects, for example – but that's another approach:
Starting at 9th level, when you fail a saving throw against a spell or effect dealing damage, you may choose to take half damage. You can’t use this feature again until you finish a long rest. You may use this feature twice between long rests starting at 13th level and three times between long rests starting at 17th level.