I am one that believes money can or at least helps attain happiness. Many of the reasons you listed are valid for how it can help, ie. lessens stress. I do have some scattered thoughts that I would like to throw in though. If you think I wrote too much, feel free to only read (IV) as I believe it is the most thought-provoking.

(I)
First of all, I don't think the phrase should be taken at face value, but more so as a moral statement. Suppose it is a preemptive statement. In other words, "money does not wash dirty hands". If someone offered you to do a deed against your own morals, ie. kill someone, for a large amount of money, would you do it? When put that way, the original statement has a new twist. No matter how much money you get, you killed someone. You won't really be happy spending that money. Of course, some do not think this way, that is why we have hitmen and dishonest leaders in this world.

(II)
For those that don't think money contributes towards happiness, I'd like to ask what makes them happy. I would pose that money helps buy time, and from time, most, although not all, things grow.

(III)
I believe money is also obviously a double-edged sword. Many get addicted to money. They can't do without it and dream about it. Everything reduces to money and it drives them crazy. All they want is more money and cross too many lines for more of it although they will never be able to spend that much in their lives.

I would say that money does not guarantee happiness, as some people seem to think in rebuttal to the claim.

(IV)
I do think this last case is the most important one, and also a very applicable one. When we attack the saying "Money does not buy happiness", our minds dart towards quick defenses, like "if I had a larger house" or "if I had a better car". However, we hyperbolize the amount of money in question. What you can do with a hundred dollars and a hundred million dollars is very different.

This leads to your life. Is an extra 3 hours at work going to be more beneficial, or will spending time with your family for 3 hours make you happier? Which will make your family happier? You may think, I need x amount of dollars to buy y for someone. But would that someone rather spend time with you over that y.

So when we bring the monetary amount closer to us instead of exploding it, there's a lot more to think about. There is a catch to the money, it isn't falling from the sky into your lap. You are sacrificing time for that money. So think about your friends, family, pets, or even yourself before you pick up another shift. Yes, money is important, but it might not be as important to them as it is to you.