Does religion tend to make people better or worse?

[All Along The Watchtower]

I think one thing religious belief can be bad for is forcing a 'disconnect' with what's playing out in front of them. For example, if there's a gay couple, together for decades, married, part of each other's families showing love, commitment, passion then to see that in play, to see it as similar to every other couple, perhaps in some cases 'better' than some relationships and to conclude; no. It's not appropriate. Not what god wants. Not real, a facsimile, shouldn't be allowed or encouraged. That's a forced disconnect. It's intentionally blinding yourself to the relationship playing out in front of you instead focusing on the sex of the individuals and your assumption of what god wants as the measure of it's truth. 'Only acts are sinful' is actually a worse fudge because falling in love and everything that entails is the 'act'.

I've struggled with working out the thought processes behind that and the effort and resilience that goes into it.

I would call that the expected result of social and cultural prejudices being sanctified and codified over thousands of years. Why some people are more "resilient" in this regard than others is a great question indeed; at the very least, it certainly has something to do with variations in social and cultural environments!

[All Along The Watchtower]

Also OP: "religions are mostly about being a better person" is categorically not true with regard to the largest, most globally dominant ones (ie. the Abrahamic ones - Christianity, Islam, and Judaism - which is not very large but certainly not irrelevant, especially when considering the roots of the two much larger Abrahamic religions...).

These religions are all based on human beings individually and collectively submitting themselves fully to God - specifically, the God of Abraham. "Being a better person" is (supposed to be) a positive consequence of that, but it's not the core of those religions, and anyone who knows anything about them and is being honest with you will tell you that - whether they are followers of said religions or not.

I hope this helps. 

[All Along The Watchtower]

Anywho, at least in Christianity (I can't speak for Judaism or Islam but as fellow Abrahamic religions I suspect  that they're similar here), humanity is inherently Fallen, estranged from God and living in a fallen world as Sinners, and so "becoming a better person" is a fool's errand. You'll never be "better." Doesn't work like that.

Accepting the God of Abraham as your Lord and following Him and His Commandments is what matters - indeed, is all that matters. Of course Christianity has a unique view of what means in terms of the figure of Jesus Christ and what Christians believe in terms of  His birth, life, teachings, miracles, death, and resurrection (FWIW, Jesus is also a major figure/prophet in Islam - in fact, I think he's mentioned in the Qu'ran more times than Mohammed, but I might be wrong there -  but is not a divine figure), but these are among the most basic things about Abrahamic religion, I think.