for the first time they are typically developing a mental illness or experiencing a notable worsening of an existing un-medicated condition.
They typically have been getting worse.
It is less than surprising that a lot of people in a downward depressive spiral both a) start taking an SSRI, and b) commit suicide.
This does not mean that there can not possibly be a bad reaction to the first weeks or months of SSRI treatment. It is possible. It means that such a link would be swamped by noise. (And hence the FDA says "may" not "can"... it may. Not known with certainty, but enough concern to be worth the warning. And the warning is of thinking about suicide, not commiting suicide.)
And we know that over a longer period of time the SSRI group will have a lower suicide rate, which is a thing worth noting when talking about whether a drug causes suicide.
On the other side of the coin, many people first see a psychiatrist when they are near the bottom. They have been getting worse, but are near the end of the worst of the thing (for the time being).
In that case the benefit of an SSRI will be exaggerated. This applies to many medical treatments. People with a condition that will improve on its own will tend to get treatment when the pain/illness is at its worst, and whatever is done will appear very efficacious. (Many patients report improvement right after beginning an SSRI, days before the drug, which has a cumulative effect, is doing much of anything. Placebos do pretty well short-term in anti-depressant trials, and perform about as well as SSRIs for moderately depressed people.)
And this is the SSRI paradox. They are not as dangerous, or as effective, as case histories imply. Anything that is typically given to people in crisis will show exaggerated effects because people in crisis have exaggerated outcomes, both good and bad.