Ask HN: How to manage shared Email inboxes?

Next to our private company email addresses, we have 2 shared email inboxes (info@...) that are being worked on by multiple people at the same time.

This is currently done in GSuite, simply by multiple people logging in at the same time. The volume of the emails in these inboxes are starting to get out of hand, and we are looking for a way of managing this differently.

The conversation view in Gmail is becoming somewhat of a hassle for certain situations, as its not always clear when a client replies to an email multiple times in a conversation flow, to what the client was replying in the first place. We tried not using conversation folding at all, but this is completely unmanageable as we are receiving too many emails a day.

I was wondering what the experiences of HN is with similar situations, Im mostly interested in self-hosted, cloud applications or addons for GSuite. As I do no want to install software on computers.