Your theory would be valid if there were no mentally ill people in those European countries which offer good access to healthcare. Just google "Magdeburg attack", if you want a recent and very good example for this.
Germany [1]: > The average waiting time from the initial consultation to the start of therapy is 142.4 days.
[1]: https://www.bptk.de/pressemitteilungen/psychisch-kranke-wart... (German article)