I use Apple Mail, with two IMAP mail servers. So all those messages are also on the servers and could be accessed via web.
Archiving the email to a folder usually means removing it from the IMAP server and storing it locally on one of the clients (2 Macs, iPhone). But you can create folders in the IMAP structure that will sync to all devices and tidy stuff up that way. Don't know if that makes any difference in terms of "load", though.
I used to be really bad at managing my email, but I adopted an inbox zero approach a couple of years ago and never looked back. I put together an article if you are interested.
I use a LOT of Mail's Smart Mailbox to organise my email. For instance, FaceBook notifications go into one Smart Mailbox. Then, I can easily select them all and delete them when they get too numerous.
In fact, I rarely look at my ACTUAL Inbox, as I have a Smart Mailbox Inbox that I use instead.
That shows Sent and Received messages for everything that doesn't go into any of the other Smart boxes. I have Smart boxes for my Uni, orders from my webshop, PayPal, eBay, various forums and software companies that I talk to, etc, etc.