But choice of method does affect rate of death, so it's correct to say that increased access to any method that's commonly used affects the rate of death. Decreasing access to that method is valid if its one of the measures taken to reduce suicide rates.
So it's less that guns increase the rate of suicide, and more that they increase the suicide rate to suicide attempt ratio due to their efficacy. Which, similar to the point you're making, means America probably has less suicide attempts to begin with than its suicide rate would indicate relative to other countries.
On the shittiest days I see the lines of nitrogen tanks we use at work for Gas Detector calibration and remember my old buddy.
[1] - https://pubmed.ncbi.nlm.nih.gov/29684846/
[2] - https://en.wikipedia.org/wiki/Inert_gas_asphyxiation#Suicide
For reference, nine of the top ten nations for suicide of banned civilian gun ownership. All of the top 40 nations make it difficult. USA is usually around #50 or so.
It is complicated, and even statements like ”access to firearms has been shown to affect suicide rates” are could be misleading because you could be putting the cart before the horse, if someone has chosen to kill themselves, it reasons they will chose a fast/painless, and effective tool.