Sometimes they can.
I will describe a case here, that goes to a believe, which while not religious, pretty much explains how it works:
I was playing a video game, some sort of a first person shooter. I reached a boss that I was unable to beat in more than 15 attempts. Got very frustrated and entered the invincibility cheat (the GOD-mode). Defeated the boss in less than 2 minutes, without taking a single hit. When I opened the console, to deactivate the cheat, I realized I have entered it with a syntax error and essentially it hasn't been active at all. So in this particular case it was not the actual invincibility cheat but just my own belief that made me invincible.
I think something like this happens with religious people too - they believe they are protected by some invisible force in exactly the same manner that I believed I am protected by the cheat code. Which makes them more confident and confidence often brings pretty good results. Which does not mean there are not other ways to get the confidence in question, while avoiding the damage of institutionalized religion.

No faith in god or religion. Faith in me, overcoming my obstacles by pure will and dumb luck.

I think that the idea of faith is the most seriously damaging thing about most religions.

What do you mean by "worthwhile" - what positive contibution to a person's psyche could it have?

It can be personally beneficial but the consequences are best described by Christopher Hitchens "Good people do good things, bad people do bad things. If you want a good person to do bad things, you make it a religion "

It made me feel powerful at the time. And you get ego-boosting adulation from the chicken-shits who live vicariously through you. And give you money. And vehicles. And want to fuck you. Iron your clothes.

Exhibiting powerful faith is mesmerising to those that don't. It's quite sick actually.

I think science has confirmed that it can have beneficial effects.

The "Placebo Effect" is well documented.