I can't answer for other atheists, but I know to be suspicious of experiences too. They also don't have to be true no matter how powerful they are and how real they seem. Been there, had some doozies. Turned out not to be what the most decidedly appeared to be on first encounter.
John Wesley found "his heart strangely warmed" and interpreted that a particular way.
But a range of things can produce a "heart, strangely warmed."
You may be right in general, I can't say. I can only answer for me.
I hold that "faith", unspecified, is at least as often vice as it is virtue.
(Wouldn't a Christian evangelist wish someone to STOP having faith... in their non-Christian deities or beliefs?)
Gymnastics no. Straight line wherever possible. Anything else gets in the way of the content of a discussion, which is far more useful.
But yes, check each link in a chain of argument, or look to see the supports of anything asserted as universal truth.
Why wouldn't you, if the issue was important enough?
Chris.