This is a tangent, but when in the world did "happiness" become a desirable metric? If you think about it, it's really quite absurd. Happiness is a brief liminal state that should be triggered by relatively infrequent events. It is not a normal, nor desirable, default state.
Contentedness, satisfaction, at-peace, and so on - there endlessly more rational, logical, desirable, and attainable things to aim for. Yet everybody always says happy. Maybe this even goes some way towards explaining the plummeting mental state of the West at large. If one sets their life goal towards happiness, then they're ironically certain to end up unhappy, unsatisfied, and discontented.
You are happy to receive good news, or for something to turn out well, or whatever else. But it is not a resting state. It's a liminal state. Contentedness, by contrast, is a resting state. You can awake contented, fall asleep contented, and spend your days contented. You may rarely, if ever, experience happiness - yet find yourself able to find satisfaction in life nonetheless.
By contrast a pitiful, depressed, self loathing individual, can experience happiness as much as anybody else. But he is most certainly not content nor satisfied. Perhaps a junky would be another example. A junky certainly experiences happiness when his poison enters his veins, yet he almost certainly is far from content or satisfied.
No, the words just have extremely different meanings. A child opens a Christmas present and starts rejoicing - nobody would claim 'Ah, look at him - he's so content!' One could even take this a step further to add that children, in general, cannot be content.
Contentedness is not happiness. And happiness is most certainly not contentedness. They're are just entirely different states of being.
First, it's important to understand that words mean what people tend to use them to mean.
Second, "happy" and "happiness" subsume numerous meanings. You're picking one, and as I and others have, at times, done with many other words, trying to restrict the world to only that meaning.
Perhaps a more similar word to your meaning would be "joy"? It seems more generally restricted to descriptions of brief periods, in common usage.
If I were simply restricting the meaning, then the examples I am offering would mean you could use happy or e.g. contentedness interchangeably, or at least doing so wouldn't sound ridiculous. Yet I'm sure you're realizing by now that you cannot! For another example - a stern and disciplined person, who is rather dour of character, could be completely contented, but it would be illogical to call this person "happy", however you might want to define the term.
The words are simply not synonyms, or even particularly close to being synonyms.
Agreed that affirmative happiness is very hard to think about as a target.
But I find that most people, when they say that, actually mean reduction of suffering. That's easier to quantify--but still quite difficult, like most quantities in social research.
Do you not then run into other problems? For instance I find that lifting brings an immense amount of contentedness, yet it's essentially hours upon hours of self inflicted suffering and pain. The same is true of family. Somebody raising a 2 year old could describe it in many ways, but reduction of suffering would not be one. Such things greatly contribute to this sense of contentedness and satisfaction.
Contentedness, satisfaction, at-peace, and so on - there endlessly more rational, logical, desirable, and attainable things to aim for. Yet everybody always says happy. Maybe this even goes some way towards explaining the plummeting mental state of the West at large. If one sets their life goal towards happiness, then they're ironically certain to end up unhappy, unsatisfied, and discontented.