Yes prayer works!! I have in fact had a lot of benefit from prayer, sometimes I’m moved so powerfully and really feel His presence that an hour long prayer session flies by!! It’s hard to adequately explain what that feels like.
But I’m afraid you might be looking at this from the wrong perspective. Prayer is not something provable in a laboratory. Anymore than asking your father to help you out with something is provable like that.
Prayer is essentially a relationship. A relationship with our creator. God is not a vending machine or a genie that grants my wishes as long as I do it right. That’s why those types of experiments prove nothing either way. He wants us to come to Him for our needs, yes, but what we think we need usually ends up not being what is truly best for us. So it’s not surprising if not every prayer is granted in exactly the way we expect.
But more than our needs, He wants us to get to know him, to really love Him. He wants to transform us, and that’s what prayer ultimately does, it takes us out of ourselves, and focuses on the Only Good in the entire universe, God. That’s when prayer really starts to change us, when we start loving Him because of who He is, not what he’s done or can do for us.
That’s why I really dislike those prayer experiments, they trivialize God and prayer. It’s rather insulting to God. It’d be like having a bunch of scientists telling me my relationship with my dad must either be horrible or fake because he doesn’t give me everything I ask for exactly how I want it. Uhhh no. Love isn’t like that.
Those scientific experiments are starting with faulty assumptions about God and prayer, don’t look to them for whether or not prayer works. Just try and pray yourself and see what happens, even just for five minutes a day. You give God an inch He’ll make it into a mile.
I recommend a book by Peter Kreeft called Prayer for Beginners. It’s a good introduction. Also try An Introduction to the Devout Live by Francis De Sales. That book changed the way I pray. Especially the meditations in the first part, if you do those sincerely, I guarantee you’ll see the entire world with a new pair of eyes!