tl;dr– Frank's arrived at an absurd conclusion, which successfully demonstrates that one-or-more of the premises/arguments used to arrive at that conclusion is faulty. However, since Frank hasn't yet demonstrated that all of the other premises/arguments are necessarily solid, it doesn't yet follow that Miles's claim is the faulty premise. Frank's argument-as-stated is a non-sequitur.
Reductio ad absurdum attacks a system of arguments rather than a specific premise.
Reductio ad absurdum is when a system of logic is applied to arrive at an inconsistency (absurdity). Doing this demonstrates that the system of logic is inconsistent (broken).
Reductio ad absurdum can be used to attack a specific argument when it's done using only that specific argument and well-accepted premises. Then when the overall system is shown to fail, the target argument can be faulted as the other arguments are assumed to be faultless.
For example, consider the argument:
If A and B, then C.
If C and D, then E.
E is contradicts F.
This shows that A, B, D, F, Claim 1, Claim 2, or/and Claim 3 is/are absurd. This can be taken to imply that A is absurd only if all of the others are eliminated as possible sources of error.
RE: The Frank-and-Miles scenario.
In this scenario, Frank has successfully demonstrated that whatever system of logic he used to arrive at his conclusion is broken. However, Frank has failed to demonstrate that Miles's premise is the faulty component; Frank's absurd conclusion may be due to another one of his premises or/and arguments being absurd.
So, as stated, Frank's argument is a non-sequitur, as the conclusion does not follow from the stated premises.
In common practice, there're two ways Miles might respond:
Dismiss Frank's argument as a non-sequitur.
As stated, Frank's argument is a non-sequitur. If Miles doesn't care to help develop it, Miles can just disengage.
Ask for Frank to explicitly lay out his arguments.
As stated, Frank's argument is a non-sequitur and thus invalid – but, this could be fixed if Frank lays out all of his arguments. If he does so, then the new statement of Frank's argument could be assessed.
If Miles can show that one of Frank's other arguments isn't solid, then Miles can demonstrate that the absurdity doesn't demonstrate the inconsistency of Miles's own argument.
If Frank can successfully argue his position using only arguments that Miles agrees with, then presumably Miles ought to accept that Frank has demonstrated an inconsistency in Miles's beliefs.
Since reductio ad absurdum isn't an attack on a specific premise, someone making a reductio ad absurdum is putting forth an attack on the collection of all arguments that they've used in their argument, as this is strictly necessary for their conclusion that the contested premise is faulty to follow.
However, in common social situations, not everyone seems to get this. Folks might put forth a reductio ad absurdum, thinking that they're attacking only the premise that they're arguing against. Then when their other arguments are requested for critical analysis, they may feel personally attacked. This might be described as sealioning.
In such situations where an arguer isn't willing to elaborate or have their argument critically assessed, then it may be socially advisable to disengage them.
I note this here because a good reductio ad absurdum can require a lot of critical analysis to sustain, to such an extent that it's often best to just ignore such arguments.
Reductio ad absurdum is unusually non-severable.
Many logical arguments have some severability. This is, small defects don't necessarily completely invalidate an argument.
1 + 1 = 2.
Humans have a left foot and a right foot.
Therefore, humans have 1+1=2 feet.
People walking down a sidewalk wear shoes.
Therefore, people walking down a sidewalk wear 2 shoes.
Premises (2) and (4) aren't necessarily true. Still, this sort of argument is severable in that, while perhaps imperfect, it's a reasonable observation that's generally approximately true.
Now let's say that Frank and Miles are chatting. Frank lays out this argument, then appends:
Look at that guy; they're not wearing any shoes!
Therefore, 1+1=2 is absurd.
While we may've tacitly accepted the earlier arguments in a general context, appreciating that the conclusion in (5) is mostly correct, Frank's argument in (7) is clearly silly.
Point being that, while reductio ad absurdum is a valid mode of argument when done correctly, it's unusually non-severable. Since it's an attack on all arguments, we have to be stricter, attacking claims that, in other modes of reasoning, may've been appreciated as approximately true.