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.
I don't think any solution will be perfect at eliminating all crime, ever. A statement doesn't have to be all or nothing to be valid, and I might also just be completely wrong. That being said it seems like a good thing, for my definition of good, to help the mentally ill.
Sure, but in that spirit you then wouldn't need to mention that healthcare access would help. There will always be those crazy enough who will not make use of access to healthcare but prefer to make a plan on how to "seek proper justice on their own", which is what OP was referring to. The inhibition of that signaling is then a more adequate solution and should be part of the repertoire (as well as offering good healthcare).
