In agreement with Michael Karnefors' comment, this question is best approached first by ignoring human civilization in its entirety, from the bottom up. Rest and relaxation are fundamentally woven into our cultures, and culture is not something that presents a set of switches you can turn on and off. Everything is interconnected, especially fundamentals like relaxation. Did you know that primitive people spend almost 50% of their day relaxing? I think that goes to show just how hard their life is in the other 50%!
Trying to remove relaxation from human society will just lead to unrealistic environments that don't capture a reader's attention. The society will feel weak and lifeless and stilted. So much of our culture is built on relaxation in ways we don't see, so those parts of culture would be unsupported without it. For example, you couldn't have people working in assembly lines like they did in the mid 1900s. That sort of expenditure of energy could only be supported because, when they went home, they got to relax (and sleep... never forget sleep).
So what would a society look without this? Your first bullet gives hints:
Entertainment (unproductive activities with the sole purpose of being pleasant or bringing relief to stress)
"Purpose" is an interesting keyword here. Purpose is not a simple concept. In philosophy, it is a word that is picked at and refined for hundreds of years, based on refinements of other word that were refined for thousands of years. It's actually a lot of fun to look at, but certainly not simple.
Such a society would need to have a driving purpose, or it would find itself faltering. Entertainment and relaxation permits us to explore purpose. Without it, it becomes hard to find new purposes. They must be given to you by the environment (i.e. the rest of the society). A society which has been at war for a few dozen generations, living in WWII style trenches, where children are born simply because they need more people to man the guns, that might suffice.
The key would be that that purpose would have to be so ironclad that it doesn't need any exploration after it has been given to you. If you are permitted to explore the purpose in your life, it trivially opens the door for entertainment and relaxation at a cellular level. After all, we do have entire feedback loops designed to manage stress at a chemical level, and relaxation is a major part of them.
Another interesting option might be that some of the society has more openness to exploration of purpose than others. The Consensus of Parts from Andromeda may be a good example. It does appear the consensus itself is permitted to explore concepts that could lead to relaxation or entertainment, but the individual elements of the consensus are not.