But have atheists tried to disprove God for more than 2 thousand years? As far as I'm aware, atheist literature presenting serious arguments just began very recently, whereas apologetical literature is very old and can be traced back to ancient Islamic and Christian scholars (and others).
Yeah, you answered your own objection. Just as one can measure the effects of dark matter, it seems possible (at least in principle) to observe the effects God would have in the physical world (for instance, since God is the creator of the world, we should expect to discover it had an absolute beginning so that it could be said to be created).
There is an endless discussion in philosophy of religion about whether religious experience is enough justification for belief in whatever is experienced. One objection is that different religious groups have experiences of their deities (or even dead relatives) that seem equally unwavering to all of them. And yet, they can't be all true since they are contradictory. Therefore, it seems one is violating one's epistemic duties when one trusts a religious experience is revealing truths. It is a crippled epistemology.
Of course, there are various responses to this line of reasoning, but I believe I can deal with them, so feel free to present it.