True. More yields or a greater efficiency in yield generation is a benefit you always want anyway. Gaining less unhappiness by achieving this goal is redundant, gaining more yields is already the benefit.
2. I would say realistic, yes. People are happier if they have jobs with good payment, enough healthy food, education and free time activities. But is it a fun mechanic? No, cause it's supposed to be a control mechanic, denying too powerful and easy ways to win the game.
I would really like to see a elementary discussion about the goals of the happiness system, cause it seems, people thinking different about it. (and only saying it should limit expansion is a very vague goal, it needs specific definition, when the happiness system works proper to do adjustments to it)
3. The fact you are going to catch as much yields anyway, your statement is true. But then you have to answer the question, which system do you want to achieve the goal of controlling overexpansion and heavy warmongering.
1. You compared Public Works to Workshops, concluding a workshop is more desirable because it also achieves happiness in the same scale as PW, while giving more yields. If so, the effect of PW needs to be twice or thrice bigger than a workshop, happiness wise, to be worthwhile. I can't opine since I haven't tested it, so I trust you when you say the effect is too low.
2. You did ask for a happiness purpose definition several times, but i'm uncertain about what you exactly mean. Could you give some examples of what you are expecting?
For me, happiness mechanics have two purposes.
First and foremost, it limits overpowered strategies such as growth, expansion and warring.
Secondarily, it punishes excessive focus, requiring the player to produce a little bit of everything. Call it development. Call it yields mix.
So there are 4 main factors adding to unhappiness (growth, expansion, warfare and underdevelopment) and when unhappiness is too high, bad things happen. So you can push any of those factors as long as the other ones are under control. And as the game progresses, the limits on what we are allowed to do become more flexible by the grace of new resources or buildings.
In this context, a public works is just a tool for those who crossed boundaries that allows them to recover faster.
Can this be done without the project? Well, yes, if we assume that every player that whenever faces happiness problems is going to build a public works, we might as well let it occur automatically.
For example, any time that a city has more unhappy people than half its size, it will riot. Every 5 turns, the city will gain a permanent -1 unhappiness, and while a city is rioting it can't build anything, except the consulate for captured cities. In practice, this means that excessive expansion will punish the players with a few production turns lost, but they will be in better position for overexpanding next time. Will you be able to time riots so they don't interfere with your strategy?
It might not be a good idea, though, for those island cities that will be unhappy for a while until they can get some buildings on it.