Failing to find evidence for something, where one reasonably expects to find evidence for it, is itself a form of evidence. For example, when they were testing the Higgs field, they had to perform tests to attempt to produce a Higgs boson, and every time they failed to do so, they were able to exclude certain mass ranges from future consideration. To put it another way, failing to find the Higgs boson at certain mass ranges was evidence that it did not fall within those mass ranges. If they had failed to find it within all expected mass ranges, that would have been pretty solid evidence against its existence (and by extension, the Higgs field).
I suspect the reasoning that strong atheists use regarding gods is something like, "we've looked for gods and failed to find them, which itself is evidence against them existing, therefore I don't believe they exist". I am not a strong atheist, so I don't know for sure, but that stands to reason. So it's not so much a positive claim as it is a stronger negative conclusion.
I agree evidence of absence is evidence to us atheists but it is not evidence to a theist. No amount of science out there is evidence to a theist their gods don't exist. They just tweek it or deny the science.
When a theist says you have no evidence that god does not exist they are asking for objective proof. This is not possible for a negative or default position. They literally want a picture of gods not existing.
I'm glad that you recognize this, and you put it into words very well. "They literally want a picture of gods not existing."
As ridiculous as that sounds, it is as close to what some Christians seem to expect when they attempt to shift the burden. In my mind, when a thing is posited, a reasonably educated person can make a fairly quick assessment on accepting or rejecting the claim. This is obviously more difficult when things are complex, like the precise calculations needed to shoot a rocket off of earth and to the moon. However, history shows it is reasonable because we did it, so something went very right. Most people have seen images of earth or they've seen globes, or both. So the claim that the earth is a sphere is obvious, and easily acceptable (although there are flat-earthers out there, sigh.)
When it comes to god claims, the challenge is in the emotional attachment and inability to separate myth from reality for many humans. Parents passed their beliefs onto their children with enough conviction and authority, along with heaps of fear, guilt, and shame to completely embed the delusion into the minds of their young, impressionable children. This is not something that can be easily shed by many. The comfort of a loving god that is taking care of their grandparents, and will reunite them is too strong. The fear of hell is too strong.
I have also done some reading on the topic of "levels of atheism", including all of the nuanced versions of agnostic and gnostic, etc. Those labels don't appeal to my inner atheist, I guess. To me, you either accept there is a god, or you reject the claims. The inbetweeners are basically undecided in my book. If you reject claims that gods actually exist, you are atheist. If you think there may be a higher power, that is something altogether different than the claims of named gods from holy books and ancient myth. Those are specific. Higher powers, or Deism, is about as close as I could get to considering an intelligence that perhaps created stuff. I don't have a good reason to consider that possibility though, so it's another rejection for me at the moment.
I have argued many times on this site that when I reject god claims, I am not
making a positive assertion that a god does not exist, I am making a strong assertion that no god has ever been shown to exist. No one seems to want me to prove that no god has ever been shown to exist. I wonder why that is? Perhaps it is because most theists realize they will never be able to show me a god. They know their god is invisible, they know it works in "mysterious ways", and they know that the problem with accepting the premise lies with my refusal to just believe. Sorry, Jet don't play that game!
If the statement "no god has ever been show to exist" is a positive assertion, then we are back where we started. We are at the place where anything can be asserted, and the burden of proof is on the person who rejects the assertion - which is patently ridiculous, and gets us nowhere.
TL;DR - I have ZERO burden of proof on god's not existing. I will never say anything more than "show me your god." I leave the tiniest crack of possibility that there is a god involved in our universe and planet - about as much room as I give for the existence of Thor - pure mythology.