As a PhD student myself, I would die on this hill, and you would lose.

Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.

Say a student is completing a problem set and is asked to prove P. If the student writes "P therefore Q therefore True since Q was proven in class. QED" then the student has committed a significant error in reasoning. If the student instead just writes "P. Q. True. QED" then they are writing the above in a more terse manner and have equally committed an error in their reasoning. If they first write that but then amend in some "iff"s between each step I'll give them full credit (assuming iffs are valid) but be slightly worried that they just learned that you need to put an "iff" in those spots to keep the teacher happy but don't understand why.

If a strategy is not guaranteed to give proof, then you need to verify afterwards that the putative proof is in fact a proof. Just as if you get a potential solution to an equation (via solving a more general equation, perhaps), you have not "solved" the original equation until you actually check that solution, even if your putative solution is the true one. If a student does not do the "check if steps are reversible" part, then they have not written a proof even if every step is reversible! That's what's lacking form their proof.

Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.

Sure, that's fine, but to say that it is an invalid method of proof is wrong.

If I want to prove A, and I prove A <=> B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!

It is a kind of backwards reasoning that students empirically screw up all the time.

Edit: so much so, that I would definitely recommend students rewrite these proofs on assignments as an exerecise to make sure it's correct. By the time they reach a math PhD they probably don't need to do that anyomre. :)

"P implies thesis (proof) hence it suffices to prove P: (proof of P)" is a completely valid proof step, and is backwards chaining.