This is one of those tools that do one thing and do it very well. Also the pricing is convenient. Congratulations to the creators.
If you want to go the open source + self hosted route, Chatwoot [1] [2] may be for you. It is more feature rich (or bloated, depending what you need) and complex to set up, defined as "Open-source live-chat, email support, omni-channel desk. An alternative to Intercom, Zendesk, Salesforce Service Cloud etc.", and it has a shared inbox feature [3].
[1] Website: https://www.chatwoot.com/
[2] Github: https://github.com/chatwoot/chatwoot