@JohnStrong that is some question!

Beyond the fact that a recent
McKinsey report highlighted the fact that we spend, globally, $1.5 trillion per year in the pursuit of happiness (which kinda means we haven't quite got it) there is also the fact that psychology and philosophy have yet to agree on a definition of what it is. Nevertheless, let's give this one a shot and we will start from your own starting point which is exercise because it also coincides with the McKinsey report's approach that took in money spent to get fit, look healthy and feel well.
If we were to ask what is happiness? A transitory state or a steady state of being the answer is both (and this where Kahneman's work and his hypothesis makes sense). Neuroscience tells us that any subjective experience (like happiness) has a correlative neurobiological state that is determined by a particular mix of neurotransmitters and the activation of specific structures and centers in the brain.
Happiness, in the brain, is neurochemically expressed as a mix of four
specific hormones: Oxytocin, Serotonin, Endorphins and Dopamine. Each one of these plays a very specific role in how they activate happiness and a sense of well-being in us and happiness itself tends to have a spectrum value depending on context and circumstances (which is why it is so hard to study definitively). I will focus on just two: Endorphins, that are released during exercise and Dopamine, necessary to motivate us into action.
Before I do that however I will also provide a more neuroscientific/neurobiological definition of happiness both as a transitory stage and a permanent state of being. We are designed to survive and the way our body does that is by maintaining a steady internal temperature, controlled breathing and blood pressure and the equitable, just-on-time delivery of essentials to cells to produce ATP (it's called
homeostasis). Everything the outside world does to us and most things we do in the outside world unbalance that state. The body (and brain) respond through adaptations that make us more efficient (i.e. stronger, faster and smarter).
Happiness as a transitory stage is experienced when we go from one state of being where we're unhappy (or in discomfort) to another where we have resolved the issue we face. Suppose you're crossing a desert (to use an exaggerated point) and you have no water. And 50 miles into your journey, when you just about reached the maximum ability you have to get by without water, you encounter water. The extreme discomfort you felt as thirst tormented you disappears and it is replaced by a sensation that you might call relief but is actually happiness because you've satisfied a dire need. This will last only until the next moment thirst strikes you.
This is the transitory stage of happiness. It is governed by dopamine, and it basically responds to accruing discomfort as a motivator which leads to an action that generates comfort and that makes us happy for a while. The $75,000 ceiling of Penn and Princeton researchers comes into this exact example. Once you have enough water to slake your thirst and not feel thirsty for a while, excess amounts of water do not make you any happier or less thirsty.
This is the same with exercise. We engage in it because we understand that the alternatives present us with bad options we are unwilling to accept (bad health, obesity, early death), the discomfort we feel during it accumulates until we stop and homeostasis begins to take place in our body giving us the "after-exercise glow effect".
Because exercise itself is a stressor and the body is primed to mitigate stressor-responses, particularly when self-administered, endorphins are also released during exercise. The most frequent self-reported euphoria and well-being on this come from runner's talking about "the
Runner's High". Endorphins are neurotransmitters released by the pituitary gland and hypothalamus in the brain. As natural hormones, they alleviate pain, lower stress, improve mood, and enhance our sense of well-being. They are pre-cursors to the release of dopamine so they work together as part of motivation by making the stressor feels less of a stressor which generates a feeling of general well-being and then dopamine allows us to feel a sense of happiness which is only further reinforced by the return of the body's processes to their baseline levels after exercise.
Since all these are transitory effects destined to wear-off the moment the exercise period is over (or shortly after) and the moment the body's internal baseline states stabilize again, can exercise make us truly happy as a long-term effect?
Having laid this foundation I will now bring up two distinct studies. The first is from 2014 and it is a
review study of factors controlling happiness. It very logically pointed out that happiness has extrinsic factors (wealth, social status, etc) and intrinsic ones of which the major one is exercise and good health. It concluded it's a 50-50 split, more or less. The second study is more recent,
dating to 2021. It used self-reporting and subjective measures of happiness but showed that people who are active (or people who got active) self-reported feeling happier in general.
Science cannot yet give us a definitive prescription that would say "exercise for so long" in order to feel good about yourself but this is what I will add: The brain has a single code running it: survive. And to do that it tries to predict the next moment on the basis that if we know what's coming we are better positioned to survive it. To do that it acts as a prediction machine that engages in the complex social, parasocial (and even, on occasion, anti-social) behaviors we observe that help us create groups, communities, societies and countries and generate civilization. All of this is incredibly complex and energy-consuming and if we are, on top of it, feeling unwell, stressed, over-anxious, insecure and fearful we tend to increase the energetic load that is carried by the body and increase stress and anxiety and reduce our overall chances of survival as a result (we make mistakes, make bad decisions, get into bad situations, engage in harmful lifestyles and activities and overall self-sabotage ourselves).
Having a body that is structurally sound and neurochemically balanced (which is, essentially, what fitness really is) gives us a sense of confidence in our self and our actions that stems from the fact that we feel a lighter energetic load in our everyday activities, (we are less stressed, less anxious, more confident and more optimistic and as a result make better choices, better decisions, engage in less self-harming activities and so on). So fitter, healthier people generally deal better with the complexity of life and, as a result, feel happier and more confident which then tends to reinforce everything else in a widening loop.
So is there a minimum of activity to achieve this? No. Do what you need to do within the means of time, space, equipment and energy you have to help you feel in control of your own body. That feeling then seeps into a sense of control of your own life and things 'magically' improve. Is there a ceiling to that? Yes, there would be. There is always a point where more exercise is counterproductive because there is not sufficient time for recovery and repair. So if, for our example, we say an hour a day of exercise leaves you feeling great and maybe there is a margin there where you can increase it to 90 minutes and still feel great, going to three hours is not going to do anything more and conceivably may reduce the feeling of well-being and control you have since your body will have aches and pains all the time and quite a lot of inflammation.
Finding that right balance is, once more, left to us as individuals.
I hope this helps answer your question and, as with everyone else here, if my answer sparks further questions just ask away.