It's true that most effects allow for saves every turn or take more than one round to enact (the basilisk's gaze, for example). While there are still some devastating effects out there, these have become rarer. That having been said, being able to save against Hold Person every round still puts your fate in the hands of...well, fate, and you can be killed before you ever get that second save.
And of course, something I've seen play out, is that multiple saves and advantage on saves does nothing to help if a save is impossible in the first place.
This isn't a complaint thread, I didn't fail some saving throw and lose a character, in fact, my recent 5e experience is as a DM, not a player. In which I saw many saving throws forced, and made, since I haven't run for anything outside of Tier 2, and most save DC's range from 10-13. There were a few DC 15's, usually for odd saves like Charisma, and one DC 20 Constitution save against poison in Hidden Shrine of Tamoachan- fortunately that one only came up due to a incautious player, and they knew how to deal with it at any rate.
However, if you have to make enough saving throws, you're going to fail one, even if you're good at it. Fighting giant rats in Sunless Citadel, I saw every one of my party's front liners succumb to disease, which prevented them from using their Hit Dice to heal.
The Chuul is a particularly nasty critter, where if you fail the save against it's tentacles, you're poisoned and paralyzed. Sure you get a save against the paralysis each turn, and try to break it's grapple if you do, but thanks to the poisoned condition, all these rolls are made at disadvantage. That's a Challenge 4.
It's hard to bring party composition or magic items into this discussion because those elements are not standard in the game. Not every party will have a Paladin or a Cleric- not every Paladin will have a great Charisma (I've met a few who basically told me "I'm here for the Smites"), not every Cleric will prepare key spells.
Not every player is going to have a Cloak of Protection, either. I don't know how these elements can be assumed by anyone, let alone the developers.
So just looking at the majority of characters, who will have at best 3 proficient saves due to Resilient, and most likely won't have all stats at 20, it seems that there is no assumption at all about saves. Some people will be able to obviate the need for saving throws, or make all the important ones, due to various factors like rolled stats, spell buffs, class abilities, or racial traits.
Some will not. Several posts in this thread indicate that this is working as intended, and it's perfectly fine- and in truth, it could be, since this is a magical game where even death is something that can be challenged.
For me though, I've played and mastered various games called Dungeons and Dragons for some time, and telling a player "it wasn't your fault, it was just bad luck- well, actually, it was your fault for not being a Paladin, lol" isn't something I enjoy.
But if I'm in the minority, that's okay, everyone is allowed to play and enjoy the game in the way they feel best. I was curious if this was a perceived problem, and I got my answers, so thank you all, even the guy who said "5e has super easy saves that have no real penalty for failing, thanks to whiny players" (I LOL'd, and got some strange looks from my roommates, thanks for that!).