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?
I don't think this is true, it certainly is not in the case of the Mind Flayer. The Mind Flayer still has a DC 15 save vs Mind Blast or be stunned and it also has a DC for extract brain.
The main difference with Mind Blast is the stun only lasts one round now and is not save again at the end of every turn.
lasses 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.
This is not the same as a high level Paladin it is not even close. There is a difference between being great at saves and being virtually immune to saves.
Example - Save against an Ancient Gold Dragon Legendary Banishment action DC24 Charisma
The best non-figher you can play - a 20th level Paladin with a 22 Charisma has a +18 and will still fail that save 25% of the time. Your Monk with proficiency in everything and a 14 Charisma will fail 75% of the time. A Champion Fighter with Indomitable left and a 10 Charsima will fail 2%.
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.
We've never had a problem with this. This is what spell casters are intended to counter in high level play, the point though is they need to take an action to do it and "rescue" an ally.
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.
We faced the same thing in Tyranny of Dragons against a White Dragon. We addressed it with two things, first my Ranger cast Magic Stone and dropped them near the fighter (the fighter actually had no missile weapons) and later in the round the Bard cast Heroism on him.
This is the kind of things that are supposed to happen and are what makes combat challenging.
Question for you: Did your party beat the Dragon? If so imagine how much easier it would have been if you did not have to worry about 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.
The same group has played many of times and it included a campaign called Call from the Deep where most of the campaign was about Mind Flayers and the same player, playing a 2014 Minotuar Fighter in that one, spent a lot of the time stunned. Worse than that though he spent several fights dominated by Mind Flayers and fighting the Party. I actually cast Fear on him at one point to make him flee from the party and stop attacking us (to be fair I caught some enemies in it to, but I mainly did it to get rid of him).
At around level 12 he actually took Resilient Wisdom so he would not be dominated every fight, and by all accounts he loved that campaign.