I've been using Front (on desktop) as my primary email app to manage personal + 10+ shared inboxes for over a year. Much of our company is on it too. It's been great to see it mature into a real enterprise player.
With Front you don't need to use email for your teams. You can live in Front without a problem. Downside is that is a bit expensive and only offers annual plans.
I'm a HUGE fan of Front. We use it at Weekend Fund for its collaborative features, most notably shared email drafts and inbox chat. We previously used Spark, which has the same feature set, but found it slow and buggy.
Front transformed our internal customer service experience. It's the ultimate inbox powerup. I'd highly recommend giving it a try if you have a high email volume, multiple channels, or both.