Ho but they have. Unless a DM put them artificially back, they no longer work.
See, but I never saw those as effecting this behavior much in the first place. Players that were problematic would just work their way around them. The only one I ever saw have any consistent impact was experience, because it was the only one that wasn't intrinsically brittle. And nothing I see in the 5e rules reduces the ability to control that.
Carrot and stick methods can have some impact on people who are doing a mild drift into behavior you don't want, but ones that are already deciding you're unreasonable? Not in any meaningful sense. I'll accept it might work some times somewhere, but enough to be a reliable tool? Afraid I can't go there.
And again, none of this has been different in games where most of that simply don't exist, in either direction. That's one reason I often think heavily D&D-centric people have a very parochial view of how to manage player problems.
Experience? Miles stones are there my friend. And many players will try to force a mile stone in the throat of a DM. (Hey we have to level up now. It has been a while...)
They could have done the same damn thing with experience in the old days. In fact I saw people try to do it. If you can't resist that pressure, why would I assume you could resist it on any of the other carrots you mentioned?
So what is truly left? Tell me because I fail to see where, as a DM with the non optional stuff in the PHB and DMG I can nudge the PC in the direction I wish them to go without going full railroad. (This is for the sake of argument, I have no problems at all as I have reintroduced most of the above myself). If I were a non experienced DM I would be hard pressed to nudge the players in the direction I would like.
This is going to sound snarky, and I really don't mean it that way, but--
Talking to people. Honestly, if talking to players about the behavior you want can't get it done, most other stuff won't either; it'll just teach them to end run you every opportunity they get. The same people who will respond positively to carrots can usually be negotiated with in other ways. The ones that can't be negotiated with will decide that they're in an adversarial role to you and no nudging will help.
And I am familiar with a lot of other system too. Some have their own tools to bring players into the line the DM wants and some don't have any. Survival alone is often the key. At the same time, D&D has a tradition of using the above examples to "encourage" players to act in certain ways or to choose some possibilities over the others. 5ed did removed these to "emulate" other RPG but did not provided other tools that these other games had to incentivize players into certain desired behaviors.
You've got an argument regarding "survival" in some cases, but that's a brute force tool at best, and many of them have nothing much beyond experience itself or in-setting problems.
Here is an example: The merchant offers you 100 gold to guard him up to Waterdeep. For 3rd level characters, this would have been good in other editions. But in 5ed, why would the characters care? The first orog they'll see will wear a nice plate mail that can be sold for a lot more. And where there's an orog, there's bound to be some more. Unless you give outrageous rewards, the players are not incentivized to act on gold anymore. And even then, at a certain point, when the players have all the gold they need, what will you use to push them toward the story you want if they have their own agenda that goes against what you had in mind? All the tools I mentioned earlier are no longer there or do not have the impact they did in earlier editions. Especially gold, magical items and experience.
But are the rewards outrageous? If the prices land such that you're correct about that plate armor, it doesn't seem so to me.
(And remember, I'm still far from sold you can't use levelling as a lever if that's really what you want. As I've noted before, gold only had any impact in OD&D to the degree it also was experience; most characters otherwise had damn-all to do with it. Magic items? Maybe, but that was going to be a hard balance to keep, and would require you to thoroughly ignore the extent treasure tables to do it with any consistency, since over time PCs would be drowning in basic magic items. Anything that was going to really impress them was going to probably cause you problems down the line).