That's not the way my parent was talking about, nor something I'm aware of.
What's very common is people getting prescribed painkillers (oxycodone, etc.) and then moving onto heroin, because they are similar drugs. But those painkillers are not psychiatric drugs.
People are generally prescribed opiates, which are not considered psychiatric but a pain reliever, and can become addicted that way. Perhaps you're thinking about benzos, like Valium or xanax, that are both psychiatric, addictive, and whose abuse/withdrawal can be extremely lethal (especially with alcohol for an overdose combination). They are similar but have different problems. Benzos are more deadly but less life-destroying. Opiates are probably prescribed too much/for too long without patient monitoring.
Even then, I vehemently disagree with you, but I at least can follow the logic.
Arguing that clozapine, diazepam, fluoxetine, etc., are gateway drugs for heroine is an argument I've never heard before. It sounds ridiculous to me, but if you have evidence to support the claim, please post.
I'm pretty sure he's thinking painkillers are classed as psychiatric drugs, which if they were would mean he was correct.
But on your point, personally I think abuse of benzodiazepines is more likely to lead to a drug like Heroin than psychedelics are. The whole concept of "gateway drugs" hasn't been proven, but the effect of benzos is a lot closer to heroin than psychedelics (anyone who has actually tried heroin might correct me on this...), so would imagine a stronger cross-over (even if not due to a gateway effect, simply down to the fact that anyone who likes downers is likely to try other downers).
What's very common is people getting prescribed painkillers (oxycodone, etc.) and then moving onto heroin, because they are similar drugs. But those painkillers are not psychiatric drugs.