Millions of immigrants over decades, safe pensions and ever increasing economic power is not being tested? We haven't had any civil wars or are even close to it. I think that nothing has happened to truly test it is actually the result of the test.
But of course we don't know what the future holds. No system is completely safe, but I feel that we have put so many safeguards against collapse into place that I can sleep well at night and that's the most important thing.
Completely eliminating privilege is bound to spark bad things. Yes, it would be awesome if everyone had the same stuff but that is not how human brains work. You cannot go completely against human nature because human nature will rebel and bring everything down with it. To be stable does not mean to give everyone the same privilege. Because people don't operate that way. Some people will always feel more privileged. So a robust system would take measures to appease both sides so they will not rebel. It's not perfect at all because both sides will be disadvantaged at some point but that is needed to go on with society.
The demographic crisis is just starting. China and Europe are both going to be hit hard in the coming decades because of the replacement rates and aging populations. If anything, mass immigration might be Europe's only option.
Completely eliminating privilege is not the same thing as everybody having the same thing. It's the same thing as everyone playing on a fair field where all costs and decisions are internalized by those who make them and not put onto others without their consent. Human nature is a spook, and doesn't actually exist. There is no universal "nature" which can be ascribed to the 7 billion persons on this planet. Each individual has different wants, interests, goals, abilities, and disabilities. When we speak of "human nature" we gloss over that.