The unique thing about guns is that they're generally more effective at suicide than the other chosen methods. It's less about the speed of access people are mentioning, but more that when someone uses a gun to commit suicide it's more likely to work than other methods.

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.