I don't think it's an intentional mechanic that some dwarves are doomed to inevitable insanity due to their personality rolls at birth. Those dwarves manage just fine during worldgen and in non-player sites. It doesn't make any sense for a 100 year old dwarf to show up to your fort feeling fine, then suddenly remember that they can't cope with life and start sliding into despair. The player inevitably ends up wondering what they're doing wrong to cause the dwarf's happiness to start declining. Anyway, incurable clinical depression just isn't a very fun mechanic.
I don't mind losing dwarves. I'm not above savescumming if I feel I've made a particularly poor decision, but I always end up building a fairly substantial burial/memorial complex sooner or later. The difference with stress is that the deaths are usually stupid and immersion-breaking.
I'd actually keep the system of traumatic experiences causing PTSD-like flashbacks and personality changes. If a dwarf sees their whole family get killed by abyssal horrors, then it makes narrative sense for them to suffer some psychological trauma. It would be nice if there were more end-points for those dwarves than insanity, exile, or "unfortunate accident" though. I like the pilgrimage (AKA. "Sod it I've had enough of this dump!") idea, but I'd make the dwarf initiate it instead of the player. Maybe they could petition to be allowed to leave the fortress, or they could just declare that they're leaving and walk out. Given the dangers in the wilderness, it would be nice if they waited to leave with a caravan.
The current system of meetings with the mayor/priests could be improved upon. Rather than just providing a small and temporary decrease in stress it could provide a means for particularly traumatised dwarves to receive a major personality change, possibly at some cost to the player. For example, a very stressed dwarf might have a religious epiphany which allows them to overcome their past trauma, but they become a devout worshipper of that god and now spend half their time praying instead of working. Or, as above, they could decide to become a missionary and petition to leave to spread word of their religion. Or a mayor with high consoler skill might be able to help them come to terms with their trauma and end the recurring negative thoughts.
The same sort of thing could even be done with the justice system and libraries. Let incarcerated dwarves have a chance to receive a major personality change, either from introspection while imprisoned, or from being visited by the captain of the guard, the mayor, or a priest. Of course the player should be able to control whether they run a nice, rehabilitative justice system with luxuriously furnished cells, or a cruel, punitive injustice system where isolation and beatings are the norm. Libraries could provide certain stressed and creatively-inclined dwarves the option of overcoming their trauma by writing about it (with the bonus of adding more interesting books to the world!). You could even add a category of knowledge dealing with psychotherapy and self-help, and allow dwarves to gain psychological benefits from reading books dealing with those subjects.
Or, hell, dwarves could just snap in more interesting ways. Maybe the dwarf who has lost their whole family becomes obsessed with mortality and starts looking for ways to bring their loved ones back from the dead. They petition for you to build a temple to a death god and the next thing you know they've got their hands on a slab, raided your graveyard, and are headed off into the wilderness to build a tower.
Honestly though, I don't mind what system gets implemented as long as the player feels like they're in control and can make meaningful choices which affect the happiness of their dwarves. Even a very gamey system would be an improvement.