I want to setup a shared inbox like "firstname.lastname@example.org" or "email@example.com" that doesn't belong to any particular person. Instead the whole team has access to it and emails can be turned into tasks or have discussion chains attached to them. I'm using G Suite "collaborative inbox" but it's just a Groups forum with some hacks.
What you're looking for is a function of a full mail server, not just a mailbox. Group mail that operates in real-time also requires some procedures for identifying who has taken responsibility for a read item so you don't have duplication of work. But, you have two primary paths to what you want.
First, you can use a real mail server (even hosted service) like Microsoft Exchange, that will allow you to set up group mailboxes with varying levels of access control per user. This would give you individual mailboxes, but also each authorized person would have visibility or control of the group mailbox. There are probably other mail servers that also do this, but you can get Exchange as a hosted platform instead of buying and maintaining server equipment.
Second, you could have a group mailbox that is routed to your CRM system. So it's not actually a mailbox per se, it's a path to your CRM and then your CRM defines how those incoming messages are routed and handled (as support tickets, as sales leads, etc.).
Your discussion about the contents of the shared mailbox should not take place inside the mailbox. Use another tool.
It sounds like the second option I offered is going to be more useful to you because your CRM will allow you to create tasks, record notes, and give anyone with authority the ability to see all the things that happen after an inbound message is received.