I wouldn't use "should," and I'll add a couple in, but that's pretty good.
If there is a real question about something in the realm of theology, one ought to ask a theologist. Believers can answer stuff about their own beliefs and understandings, but can't be considered authoritative on doctrinal or exposition issues. It is perfectly fine to reject apologetic arguments, so long as the objection meets the standards in play. So yes, just as the creationist's "crocoduck" isn't a real argument against evolution, the Israelite's failure against iron chariots isn't a good argument against God.
Contradiction, in and of itself, doesn't defeat a position - it exposes where the situation needs clarification. One wouldn't reject physics because light is both a particle and a wave. Our root understanding is that contradictions merely tag the limits of how we understand things, a criticism of us, not the natural (or the divine).
One example of the "contradiction technique" is to apply bible literalism and point out mismatches. But accepting the bible as literal history means those mismatches are flaws in understanding what's written - the literal truth is a given in this construction. On the other hand, claiming it's not actually true by way of contradiction, only gets you to "what should I take as true and what not?" Which is what theologians argue about anyhow.
Finally, if an argument starts with "There is no God" then what's to talk about? If I claim there are no stars and everything we see out there is an illusion, what's the point of discussing astrophysics?
Okay, but the problem is that, general, the two options I listed are incompatible. What the general believers believe often has little to nothing in common with what the theologians may argue. Even in the Catholic Church, many parishoners have beliefs that are contrary to Catholic dogma. When you get into the Protestant faiths: Baptist, Methodist, Pentecostal, and so forth, it's rare to find even two ministers who agree on everything. The focus on a personal understandiong of Jesus means that, while there may be dogma, it's generally just paid lip service, and as long as everyone attends and doesn't rock the boat they can believe whatever they want in the details.
I do agree that if one is arguing against a religion in general, then you should take on the expert arguments. However, most of the arugments are with individual believers, and with the beliefs held by a majority of followers, and these rarely have anything in common with theologians.
I think this entire series of arguments ios not a category error, but a domain error. I argue at the level of most followers, not the .1% that are theologians with sophisticated arguments. However, if in a discussion with that .1%, I would address the arguments they are making.
But telling me "You're doing it wrong" when I'm making arguments aimed at the church memeber level, because I'm not addressing what Bovifeceal Aromatis wrote in the 16th century, or the pronouncemants of Pope John Billy Jim Bob Jones the 3rd (call me "Bubba"), is completely missing the point.
ETA: One other note. The wave/particle duality in physics is not a contradiciton, but a conceptual failure. Light has properties of both particles and waves, but there are no conflicts between it's observed properties. In religion, however, the contradictions aren't of the same nature. Not all of them rely on literalism, in the first place. In any case, as soon as one starts making decisions on which parts of the bible are allegory and which are literal, based on their personal interpretation, then they've already admitted that they follow their own moral code, not a God-given one. Not to mention that people tend to join a church they agree with (because it already fits with their personal morality).
There is much truth in the old adage: the only difference between a theist and an atheist is that the atheist disbelieves in one more god than the theist.