When arguments are delivered in parallel, their combined strength is no less than the strength of the strongest argument. For example, you might encounter two different people arguing in favor of the same position, one giving an argument that you can easily see doesn't work, and one giving an argument that's harder to refute. Or somebody might give five examples of a phenomenon they say exists, two of which are stretches but three of which work out all right. In both cases, the situation should have at least as much "persuasion power" as it would if the worse arguments did not exist.
When arguments are delivered in series, their combined strength is no greater than the strength of the weakest link. Somebody might begin with premise A, use it to draw intermediate conclusion B, and then use that to draw final conclusion C. If the link from B to C is suspect, then the entire argument is suspect, even if the link from A to B is irrefutable.
I'm guessing most people who will read this post already understand what I'm saying, but I've never seen it presented with the "in parallel" and "in series" phrasing before. I thought it was cute, and maybe useful, so I decided to post it. As usual, be careful not to apply this principle selectively -- your own in-series arguments need complete inspection to be accepted, and others' in-parallel arguments need complete inspection to be rejected.
Caveat: When I was getting feedback on this post, somebody pointed out that the presence of two bad examples actually can weaken the case made for the theory if your reasoning process involves evaluating how competent the person presenting it is. If they were 3 for 3 on giving examples, you might trust them more and therefore give more credence to their claims, whereas if they were only 3 for 5, you might trust them less and therefore give less credence. I agree with this. I think the reason it doesn't quite work out perfectly is that, if you're trying to evaluate the competence of the other person, then the five examples they give aren't actually quite parallel arguments, since they also contribute to the shared pool of "how trustworthy is this person?".
But really, the important point about parallel arguments is that you can't pick one of the two bad examples, argue that it's bad, and then ride the momentum from that rejection into a rejection of the whole theory. By the same token, you can't reject a position on the grounds that you thought of a reason why somebody might believe it, and the reason was obviously stupid. The problem with doing that is, that reason exists in parallel alongside an infinitude of other possible reasons to believe the position, and arguments in parallel can't be rejected collectively just because you found a weak argument among them.
An agent wanting to search for truth with bounded resources is better off looking somewhere that isn't a known cesspit of bad evidence presented as if it were good.
That aside, I still don't think your hypothetical world captures the essence of the proposal. The Gish Gallop is primarily an asymmetric resource exhaustion strategy. Evaluating the first(*) couple of arguments to judge the quality of the evidence is a defense against that strategy. If they're only presenting their own two arguments then this isn't a Gish Gallop, and the defense is inapplicable.
It seems to me that a better strategy would be to first quickly check that they do in fact already know the standard not-obviously-wrong arguments or other evidence, and then proceed with your own information. If they don't already know it, then presenting the well known information first is a good thing. If they do know it, then you now both know that you're working from a common base before starting on the new information of unknown value. In both cases your situation is stronger.
(*) They need not be the first strictly, but shouldn't be cherry-picked. Reliably choosing the first few makes it obvious that they're not cherry-picked, and establishes a useful norm of communicating strongest information first.