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I would not, as a DM, call for some of these rolls to be saving throws. According to the rules, "A saving throw — also called a save — represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm." As a DM, I don't call for  a  saving throw to "find things out" or "succeed at doing something" but rather, to avoid being affected by something negative. In the above examples, the character is not generally attempting to resist anything nor at any sort of risk of harm or negative action. The character is trying to attempt to do something proactively. This suggests an ability check, with or without appropriate skill proficiencies.

• Ruler giving a speech, you want to detect from body language if he is lying - that suggests a Wisdom (Insight) check. Gaining clues from body language, etc., is an example in the rules for Insight
• You go to the archives to do research - that suggests an Intelligence (investigation) check. It's one of the examples (poring through scrolls looking for clues).
• The guard holding back the crowd is, as said in the OP, a Strength check, not a saving throw.

With the DEX and CON examples I would call for saves. The rest, I would call for a check and not a saving throw. There is a difference -- saving throws don't allow proficiency as a general rule, and some characters get bonuses to saves but not ability checks and vice versa (eg. some PCs get advantage to saves vs. charm or sleep). It's important to do the correct thing (check vs. save) or else you could end up with incorrect results based on the bonuses/penalties each character has.

 Precisely. Many DM's conflate saves and ability checks. The Wisdom save the OP suggested should be at best an Insight Ability check.

 Or maybe passive insight.

 How would this play out though? If some or all pass, ok, they realize something fishy is going on. But what happens if everyone fails? Does this mean those PCs are not allowed to be suspicious? That’d arguably be incredibly powerful, possibly even gamebreakingly so, and it’d feel like a massive railroad mechanic by taking away player choice. But if they are still allowed to be suspicious, they just don’t necessarily realize there’s shenanigans afoot, why even have a roll (save or check, doesn’t matter)? 

To me this is starting to look a lot like a situation where the DM should just make up their mind about whether they want some or all PCs to understand what’s going on or not. That’d largely be what passive Insight to oppose the deception would effectively be anyway, in practice. Assuming we’re not going with the broken “you’re not allowed to disbelieve period” option, is it important to leave it up to chance who is suspicious? And is that even in sync with how the game usually works, given that monster stat blocks usually have fixed DCs for this kind of thing rather than put up a roll?

I can certainly see why you might say the PCs should have a chance to succeed (and, on the flip side, to fail) mechanically so it’s not just up to the players deciding how their characters feel about what’s going on, but I’d personally be inclined not to make this overly complicated (especially since for consistency’s sake you’d have to do this every time somebody decides to lie to the party).

 Success or failure of dice rolls should never dictate a player character's behavior unless there's mind control involved.  Failing an insight check just means that a PC can't tell whether or not someone is lying, it doesn't force the PC to automatically believe them.

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The ruler lays out a detailed plot he says the men on the gallows attempted against him. You listen to his story, but you know several of the events in it are extremely statistically unlikely. You go to the archives and research. The events he described could not have happened. You have made an Int save.

This is more of an Int check.  You don't make saving throws just to point out flaws in someone's argument.

 

I was thinking this too, particularly to the extent it involves going to the archives, which is an active step. That said, the classic example is spotting a flaw in an illusion, which is not that different from 'elements of the story make no logical sense.'

 

 

 Yeah, really the save, if any, is the part where you recognize the suspicious elements of the argument. You have not only the knowledge, but the presence of mind to be skeptical of all the facts. Checking the archives is something you do later, after you've succeeded on the check to avoid being immediately persuaded.

 Still wouldn't be a saving throw, much less an intelligence save.  INT save is for something like a character recognizing something suspicious about the floor ahead of them and realizing that it's an illusion spell that's concealing a pit trap.