to be fair, logic and inference are different things. logic helps you build structure around a given set of axioms.

actually pure logic can be horribly unhelpful if you start with axioms that are not aligned with reality. that's why some really smart people can believe in stupid things.

there is literally no way to use logic to infer existence of god.

Agreed

people believe in religion because they were brainwashed to it

It's way more complicated than that. First, most people normally absorb religion from our immediate environment. The biggest predictor for if you'll grow up to be your parents' religion is whether your parents displayed religious behavior, especially some that came at some kind of cost to them and you could perceive as a sacrifice (for example, if your religion says you shouldn't drink, but your father drinks, you probably won't find anything else convincing, but if you know your dad would otherwise drink, and that all his friends drink, seeing him drink water or soda when surrounded by drinkers is powerful).

Second, it's true that people who are already religious tend to tell themselves that they believe out of logic, and some of them eventually try to be intellectually honest and wrestle with these questions, but belief is only the biggest aspect of religion in some branches of Christianity.

In most religions out there, you can be agnostic and still practice religion, and the belief based model is extremely inaccurate at understanding and explaining those other religions, because unlike mainstream Protestant Christianity, which takes the belief in Jesus and the Bible as the one thing that separates a Christian from a non Christian, in other communities it can be more important whether you participate in holidays (like festivals with animal sacrifices in pre-Christian Europe; you can find evidence of Romans who didn't believe in their Gods back in the day, but unless they refused to engage in festivities people generally didn't care). There are also religions where belief is important but practice is equally important, like most of Islam.

I had religious trauma growing up and it's shit, and although I'm aware of people who had it way, way worse than me, to this day I still struggle with purity culture over a decade after formally leaving the church.

All of this is to say, brainwashing rarely works to make people stick around, people who stay due to coercion, often do because their religion functions as a high control group (such as Mormons and other branches of Christianity that isolate you from everyone who doesn't belong to your exact same religion), not because they were taught wrong information. There are exceptions, like people who are terrified of hellfire, but again, that's more often than not due to lots of abuse rather than just being told hell is real (even being told that as a child is fucked up, especially if you have OCD or intrusive thoughts in general).

So yeah, it's complicated.

there is literally no way to use logic to infer existence of god.

I suggest you read Spinoza.

We know this now, due to in large part very smart people trying to prove it can be done in scientific ways with real philosophical and logic systems, and failing.... Not just non-believers saying it's obvious.

there is literally no way to use logic to infer existence of god.

I just gave an example of someone doing precisely that. Read Descartes first three meditations. I won't say I agree with Descartes; I think his logic is flawed. But he does technically use logic to infer the existence of God. That's why I used Descartes as an example.