You cannot measure or observe what is not there.
True, but first you have to prove "it is not there". Proving something does not exist in a , supposedly but certainly effectively, infinite system (such as a universe) is not possible without exploring, classifying and measuring every characteristic of everything in the universe, or even universes. Only after that can we produce a definitive list of what is and is not there.
You cannot measure or observe God.
With our current senses and sensors.
So God does not exist.
In a physical sense so far as we are able to detect any influence on the currently observable universe.
Those that try to measure or observe God disagree among themselves with the validity of their claims on which God has acted. So they cannot measure or observe God because God does not exist.
That's their bag, leave it up to them.
So, no measurable evidence of god or any gods, nothing that does not have an explaination in the physical world.
But, if "god" is a mental concept in the minds of some people that modifies their behaviour, for good or bad, is that concept any the less non-existent than love, schizophrenia, depression etc - IIRC all these , including religiosity, are measurable in some way, if only via fMRI.
Thus religion exists and has an effect on this universe, for some the core of that religion is an otherwise imneasurable entity, focus, source or whatever. Once, apart from the profoundity of its affect on the individual, mental illness was not measurable but still considered to exist.
Is "god" a product of induced mental illness? Does mental illness exist? There it might seem "nature" is as important as "existence".
If a mental concept affects the universe as we experience it can be said to "exist" . . .