The value can be small (or nonexistent), but I like to know that everything in my inbox needs to be acted on/processed, rather than having to determine whether I've already done it or not. You can do this while leaving everything in the inbox by toggling the read/unread tags, but I'd rather not.
Similarly, I've never seen the advantage of leaving old emails in my inbox rather than just archiving everything without labeling it.
The problem I have when I've used that strategy is that then I need to skim all of the emails and recall the state of each one. I find it to be both distracting and stressful—I'm reminded of every task I need to accomplish (or, at least, the most recent 20) whenever I check my email.
For my work email, I try to practice inbox-0. Every message either gets an immediate action, or it's filed away as a task on my todo list (including the URL for the message in Gmail), and then archived.
Similarly, I've never seen the advantage of leaving old emails in my inbox rather than just archiving everything without labeling it.