Then why not just get rid of saves?
This may not be an issue in your game, but it definitely was in ours and in the games I DM I am simply going to housrule it so it is not an issue. That is an easier fix IMO.
Now I've heard that part of the monster redesign was to have more monster abilities that don't even allow saves, like how the new Mind Flayer just automatically grapples and stuns you if it hits with it's Tentacles attack. Going from this, it seems like not much has changed from 2014. Sure, maybe the Fighter gets more ability make saves, but is that as impactful if less things allow for saving throws?
As for whether or not a Fighter being a saving throw machine matters, well, I'll admit, nobody is playing a Fighter in the 5e game I run, and in the one I'm playing in, one guy has 2 Fighter levels.
Classes that make saves often already exist, like Paladins, but the problem I see more is enemies not being able to make saving throws against PC casters. Many monsters aren't even proficient in saving throws (at least in 2014), and I've had encounters where I just had to throw up my hands and surrender because I keep failing saving throws turn after turn.
I find it more of a problem as a player that there are saving throws that most characters will never be able to make, no matter how hard they try. I find that a problem in design that gets worse as the game progresses. Even with Resilient, many classes have three terrible saving throws, and even if you throw all your ASI's into buffing your off stats, it's not going to help much.
In Storm King's Thunder, we ended up facing a powerful dragon when my Fighter could only make a save against it's Frightful Presence on a 18. I did not, as one might imagine, roll an 18, lol. This left me unable to approach the creature, and all my ranged attacks were made at disadvantage. The Cleric was far too busy healing the party to get a spell off to help me.
And ironically, the next time that came up, saving throws did, in fact, no longer matter, because we started using our massive amounts of gold to pay for the Cleric to cast Heroes' Feast each day, making me flat-out immune to being Frightened.
I'm sorry if you felt frustrated that the Fighter in your group never had to worry about making a clutch save. I'm sorry if that made the game less fun for him (assuming he felt like he had no weaknesses). But I wonder how he'd have felt if he'd been hit by a no-save stun from a Mind Flayer. Would he have enjoyed the game more knowing that sometimes, nothing you do matters, and you can be handed an L out of nowhere, through no fault of your own? I know some people like playing on the razor's edge, with that sort of risk. Rogue-likes exist, after all. People love them some Dark Souls or old school dungeon crawls where failing to poke everything with 11' poles and actually engaging in combats can lead to a pile of crumpled up character sheets.
I played that way for many years. Then I got bored of feeling like I should start naming my characters "Bob #'s 1-15" and not bothering to give them real personalities, because their existence was so fleeting. Which isn't an indictment against anyone's preferred play- it's just not what I enjoy anymore.
And until they tell me otherwise, I don't think anyone I play with enjoys that either. Though I could be wrong, and they secretly shake their fists at me every time I fail to grind their characters into tomato paste.