Well technically you can mess with servers and rerouting and such to access the mail from multiple computers, but it's a real pain in the butt and there are always a ton of problems and weird errors that come up all the time. You can set up separate POP mail accounts on multiple computers, but if you save an email in a folder or mark an email as read on one computer, it won't show up as read on the other nor will it be moved to the folder. The only way to make everything universally update on both computers is to set up a centralized server, and that's way out of most people's league. Even I wouldn't want to do that.
As for having all your email in one place, it's much easier to just have a different email in each tab in a browser. For example, right now I have my Gmail and GoDaddy email accounts up in this browser. All I need to do is click each tab to check my emails. I suppose if you've got like 10 different Gmail accounts it would be a problem because you can only be logged into one Gmail account at a time per browser, but even if you have more than one Gmail account, just have one open in Firefox and one in Chrome or whatever browsers you prefer.
I'm not trying to turn this into a debate or argument. I'm just pointing out that email clients really don't have many advantages for regular everyday people.