Morphene
I'm not suggesting that infinite recursion implies the need for a God - i'm simply saying that infinite recursion is counter-intuitive to the everyday experience that indicates that all coherent experiences require a principle that explains them. As no naturally observable principle can be self-sufficient, it is at the very least plausible, and thus at least can be considered reasonable, to believe that there is a self-sufficient principle which underlies what we can experience. What I am hoping that addresses is the first potential objection: people who believe in God are fundamentally unreasonable or irrational (as others, though not yourself have suggested). I hope that with that you can consider that at least believing in God is not a fundamentally unreasonable position, and that people can find it a perfectly reasonable and plausible position to hold, even if you disagree yourself.
That said, if your main issue is with the particular religion that one adhere's too via whichever particular revealed theology they believe then that requires a great deal more understanding. I can only speak from my own perspective here, which is Roman Catholic, although I would hazard that I could mount a reasonable defence of most Protestant theology as well (even if I disagree with it!). I am far, far less au fait with non-Christian religions.
As for the notion of supporting an institution which is corrupt, then I think that you and I would generally agree. Corrupt and immoral behaviour degrades both the individual and the institution to which they belong. As to whether or not the actual set of beliefs that institution holds necessarily means such corruption is a part of it, or if corruption is simply a reflection of the human condition that affects all; sinners and saints alike, then I think it's a different question.
And as for your disconnect about people saying 'Thank God for their recovery', I think that unless you believe that there is a revealed God who cares for and loves His creation, then yes, it would seem anathema to thank a God who you do not consider exists. I am not saying here that I think God is solely and 100% responsible for the recovery of all persons - medical staff and technology of course have a great deal to do with it, but that does not mean that the final call to death does not also have something to do with God. That is however something I imagine we are unlikely to agree on.
That said - is it such a crucial issue that you believe the friendships which have developed have to be cooled or soured over it? Surely it is possible to be friends with religious people, or even Brexiteers or conservatives? They may not be your best friends in the whole world, but that doesn't mean you can't maintain a reasonable discourse with them, to share your hopes and dreams and thoughts and feelings without expecting to be trashed or laughed or dismissed? If you felt that was likely then of course I would not want to be friends with such people either; I am more concerned that the people I choose to be friends with are people that I can share the most intimate parts of my life with; my wife foremost among them, but a few others too. If i felt I was to be mocked for who I was, then I would cease to be friends with those people as well. Otherwise, perhaps these people are people you can be friends with, good friends, loyal, kind and thoughtful people who will love you and support you, even if you have different ideas about who or what originated the universe?