I don't think that "etiquette", per se, is a real issue. If the previous author's result was published, it was no doubt considered to be a useful contribution, even with the proviso (conditional). You don't have to remind everyone that it's OK to publish such results.

Simply state the facts: "Smith [1] provided a proof of theorem X under the assumption that Y is true. Here, we present a new proof without the necessity of relying on theorem Y"

I don't think that "etiquette", per se, is a real issue. If the previous author's result was published, it was no doubt considered to be a useful contribution, even with the proviso (conditional). You don't have to remind everyone that it's OK to publish such results.

Simply state the facts: "Smith 1 provided a proof of theorem X under the assumption that Y is true. Here, we present a new proof without the necessity of relying on theorem Y"

As to the issue about how the theorem is named, there can be quite a bit of variability. Eponymous theorems or hypotheses (Pythagoras's theorem, the Riemann Hypothesis, etc.) compete with more descriptively named theorems (the prime number theorem, the convolution theorem, etc). Early proofs of the prime number theorem rely, to my knowledge, on complex analysis, whereas the proofs by Atle Selberg and Paul Erdős are "elementary" (in the way that only mathematicians can sensibly use that word) ... but the theorem is still known as the prime number theorem. It is likely that the system of referring to the result, rather than the method of proof, will still hold, whether the later simplification is doing away with an underlying conditional (as you propose), or changing the approach altogether.

I don't think that "etiquette", per se, is a real issue. If the previous author's result was published, it was no doubt considered to be a useful contribution, even with the proviso (conditional). You don't have to remind everyone that it's OK to publish such results.

Simply state the facts: "Smith [1] provided a proof of theorem X under the assumption that Y is true (or false as the case may be). Here, we present a new proof without the necessity of relying on theorem Y"

I don't think that "etiquette", per se, is a real issue. If the previous author's result was published, it was no doubt considered to be a useful contribution, even with the proviso (conditional). You don't have to remind everyone that it's OK to publish such results.

Simply state the facts: "Smith [1] provided a proof of theorem X under the assumption that Y is true. Here, we present a new proof without the necessity of relying on theorem Y"