I can empathise with you, as I have a similar experience which I will briefly relate. I had a very good friend from childhood who was a complex person who struggled in social groups, but one-to-one we understood each other and had a great friendship. We were solid friends from age 9 until about 18, whereupon we fell out of touch, mainly because I had progressed to associating with friends in groups and this friend just did not have any desire to be with groups of people. After 20 years without contact, I attempted to contact him again - we spoke on the phone once, but it was clear we were now completely different people, and while I tried very hard on that phonecall to bridge any differences, he was utterly uninterested in rekindling any friendship.
You don't mention your age, but if you have known Alex for years then it may be that these 'toxic' personality traits you mention were not as apparent when you were younger. To quote Robyn Hitchcock, "uncorrected personality traits that seem whimsical in a child may prove to be ugly in a fully grown adult".
Like many objectionable personality traits, there is a difference between someone who is passive-aggressive, and someone who has got into a habit of using passive-aggressive behaviour to get what they want. Hopefully, you can influence him in a positive way that might make him reign in these traits and achieve your stated purpose; however, this article on passive-aggression from Psychology Today says that efforts to change people "often end in frustration and disappointment", so you should perhaps also prepare yourself for the outcome that he will not change, and that he is just never going to fit into your group dynamic.
On the subject of how to deal with passive aggressive behaviour, the behavioural psychology journal VeryWellMind.com says:
"When the other person begins acting in such a way, try to keep your anger in check. Instead, point out the other person's feelings in a way that is non-judgmental yet factual".
Following this expert advice, and using the specific example you gave in your question, calmly confront Alex with the essence of what you believe he is angry with. Perhaps say:
You seem to be angry/annoyed at [other friend's name] talking about their problem.
And then ask:
Is there a reason for that?
You could either address this behaviour as it arises, but this may cause some embarrassment in front of your other friends. It might be better to retrospectively address it with Alex privately.
You should be prepared for Alex to deny the behaviour, or that he is angry. The same article says that such persons may well "[deny] what they are feeling and [refuse] to be emotionally open... shutting down further communication and refusing to discuss the issue". Ultimately, let him know that his behaviour causes ill-feeling among the entire group including yourself, and ask if he can moderate it in future. Tell him the effect that such behaviour has on the others.
The resource also suggests that one reason why passive-aggressive behaviour develops is in family situations where people have not been allowed to be emotionally open. The second part of your objective, to minimise the clashes between him and your friends, might be to help the others understand him a bit better, as they do not have the benefit of knowing him as long as you have. Perhaps give them some detail of how he can be, and why he may be like that - without stepping over boundaries of confidentiality.
Ultimately, the only people we can and should control is ourselves. You can hope to influence Alex's behaviour in a positive way, but you cannot control it. Your aim should be to highlight his toxic behaviour and its long-term consequences. If he hopes to retain your friendship and those within your peer group, he will need to make changes within himself.