For the most part true, but the "not thinking rationally" part is a poor choice of words to say the least. What they mean by "their mind isn't functioning correctly" is that suicidal people aren't behaving like they're told to.
There's the obvious attempt of telling those formerly close to the deceased that the suicide wasn't their fault so that they (and society) don't have to take an uncomfortable introspective look at their actions and beliefs that contributed to the suicide.
Then they dismiss all the pain, contemplation, and internal conflict the deceased had endured by acting as if it was some embarrasing drunken kiss at a frat party in college, and essentially end their charade by saying "they made bad choices". Because people CHOOSE to kill themselves, so that's their problem and I (all parties involved) don't have to change and should be comforted for MY loss [of comfort].
Compare these two lines from the text notice the shift in tone from sympathetic and understanding to judgemental and angry.
P1: "When I have spoken to people contemplating suicide they are usually in so much pain and despair they're not thinking of anything except stopping it."
Notice the choice of words "they're not thinking of anything" rather than "they can't think of anything". Implying that this "pain and despair" isn't so great that it wouldn't bother them if they were "normal", which goes back to the "only the weak-minded get depressed" mentality.
P4: "... People who leave us during depression or mental illness are not thinking at all. The suicide is a horrible symptom of their mental illness"
They explicitly restate that part they had carefully hinted at earlier, feeling safe enough after reassuring those close to the deceased that their hands are clean. Then claims that this suicide (possibly all) was caused by mental illness, which proponents always claim is either: A.) Genetics, (inferior being) B.) Degenerate lifestyle (made bad choices), or C.) A "bad apple" (wouldn't have happened if it was someone else under same conditions).
They close it by implying suicide is "a cry for help", that he probably wasn't serious and just wanted attention. Suicide is a result, not a symptom. This person was probably unaware of anything they said, just regurgitating the bullshit they heard from psych 101 or online.