Question:
How can (weak) atheism be automatically justified when it doesn’t offer an answer to the biggest question any of us will ever ask: does a god exist or not?
Response:
In principle, atheism should be automatically justified in the absence of compelling arguments from theists on behalf of the existence of the god they claim exists. This is because without such arguments, the default position should be to not bother believing in it — just as is the case with any other possible claim in any other subject. Some religious theists have trouble with this because they don't treat the existence of their god to be very like any other claims which people make.
Do Gods Even Matter?
For most religious theists, the existence and nature of their god are the most important things in their lives. Their religions are entirely centered around what this god wants, what this god has said, and how humans should relate to this god. This is why they treat the existence of their god as the "biggest question" anyone can every ask. It is, at least, the biggest question they've ever had to ask and answer. Atheists, though, don't necessarily grant this as true — at least for them.
For irreligious atheists, the situation looks quite different. Because atheists don't grant that any gods even exist, much less that they have any particular nature or demands, then the mere existence of some alleged god is not necessarily the "biggest question" we can possibly look at. Maybe there exists a god that is important, but maybe there exists a god that isn't important. Atheists cannot automatically grant the theists' premise that the existence of their god matters.
Atheists as Skeptics
Another important factor is that atheists typically try to use a skeptical, critical approach to extraordinary claims — including claims about existence of gods. Atheists aren't going to privilege this particular issue by using lower standards of proof, evidence, or logic. If theists wish to offer arguments on behalf of the existence of their god, they are expected to adhere to the same standards as anyone else making dramatic claims that appear to contradict what is known in the physical sciences.
It's not uncommon for believers in other paranormal or supernatural things to also be convinced that their beliefs are (or should be) supremely important for everyone. Believers in Bigfoot or alien abductions sometimes wonder why others aren't as concerned (or obsessed) with the issue as they are. Skeptics, though, don't privilege their claims as being inherently significant just because the claimant thinks those claims are important. It's their job to demonstrate that they are on to something others should care about.
Atheism Isn't an Ideology with Answers
Finally, there is the simple fact that atheism is not a philosophy, ideology, worldview, belief system, or anything similar. Atheism is merely the absence of belief in gods; as such, it doesn't offer any answers to any questions and cannot be expected to. The absence of belief in elves or angels also doesn't offer any answers to any questions, but no one expects them to so no one tries to criticize such positions on this basis. No one alleges that disbelief in elves is dishonest because it doesn't answer the question of whether elves exist — that would be absurd.
Atheism does not answer the question of any god's existence because atheism is simply the state of not believing in any gods. That's it, nothing more. If someone is looking for answers, they should look to philosophies or belief systems which address the relevant issues. It's not up to atheists to help theists by making their arguments for them; it's up to theists to justify the claims they are publicly making.
