"I sat through a service a few years ago where the opening prayer encouraged the congregation to ‘reflect on the times this week where you haven’t lived as God would want you to, where you’ve disappointed him and let him down’. It’s basically a big guilt trip, mostly devised by men to control women."
But you've missed the bit that comes next, where, having reflected on those times, you hand them to God and He takes all those 'failures' away and, from that moment, you start again, as free from your mistakes and failures as you were the day you were born. Now you are free to enjoy being free and forgiven and try to live as you believe God wants you to. The point being that you don't need to feel any guilt once you've acknowledged it and repented because your past sin/failure is taken away.
Unfortunately it doesn't seem to be that easy and you let God, and yourself, down again so you repent and hand the junk over to God again and again. And start afresh over and over, in the hope (a belief-based hope) that it'll get easier (with His help if you believe).
That's the idea. I would have thought it would be pretty meaningless if you don't believe that Jesus died taking all our sin with Him. And even more meaningless if you don't believe that you sin or that there's a God to let down.