Thanks Brian and others for suggesting this feature!
We are considering to add a setting to hide all read channels in the sidebar: https://mattermost.atlassian.net/browse/MM-7502
In the meantime, there are two features that can help your teams:
1) Mattermost v4.7+ supports a setting to group all unread channels at the top. This makes it easier to find all your unreads. It must first be enabled by your System Admin: https://docs.mattermost.com/administration/config-settings.html#group-unread-channels-experimental
2) Mattermost v4.9+ shows unread channels when using the channel switcher (CMD/CTRL+K)