Gmail – Automatically Delete Old Messages with Specific Labels

I have a ton of filters created that label e-mails in my inbox as they come in, so that I can easily discern between what groups they belong to with a quick glimpse. There are e-mails that get nested under, e.g. LABEL, and these are e-mails I read once and then archive, or sometimes just mark as read and archive (the subject line sometimes is enough to know what's going on inside it). However, after a week or so, the messages labelled LABEL are no longer of interest to me, and I delete them to prevent piling up of read and archived useless messages/e-mails.

What I want to know, is if there is some automatic way to get this latter part accomplished, since I now need to delete e-mails older than 7 days nested under LABEL manually. I know I can use filters to, in addition to adding a label to them as they come in, also delete or archive the e-mails. But that's not what I want; I want to be able to accomplish something like this:

1. E-mail comes in, filter acts upon it, and I see it labelled as LABEL as it drops into my inbox. This part is why a filter (as far as I could tell) is of no use for what I want, since I don't want the e-mail to skip the inbox;
2. I either read it or mark it as read, and then manually archive it;
3. It goes away from my inbox, but I can still access it through search, All Mail or LABEL;
4. It gets deleted 7 days afterwards, automatically.

Is there anyway to accomplish this? Either via the tools GMail has by default, or by some other means?