Respirator is the correct and less-confusing term for the item you're referring to.
Everyone should probably wear a surgical mask, because it will help contain the virus to those who have it, and help keep it from spreading. That's what masks are designed to do.
Wearing a respirator is overkill for most people, especially for everyday use, and therefore should be avoided because of the supply problems. They also have the usage complexities you mention. The fit tests verify fit, and are used help choose a well-fitting mask. I'm not sure how often the tests force someone to choose a different type.
I'm not convinced respirators are useless for regular people though, if you're a high-risk person in a high-risk area (such as an elderly person visiting an ER), I think their use might be warranted (coupled with more important practices like strict hand hygiene, etc).
Even if the mask isn't perfect in protecting you, it's probably pretty good for protecting others when/if you are sick.
Containing your own viral spread is as important as protecting yourself when you think at the societal level...
Moreover, it's significantly fewer people who are all sick at the same time. This means more available hospital beds. It means more time to develop better treatments. Less pressure on supply chains. Less societal impact from people who can't work.
It's at best marginally effective at reducing a person's probability of catching or spreading a disease. But in the face of potentially exponential growth, reducing that base by a few percentage points is a huge win.
That's why vaccines are incredibly effective despite being only 90% effective. Disease simply doesn't spread when 90% of the population is immune, no matter how virulent the disease is.