If you're looking for a unified interface, perhaps Opera is the answer. Its mail client (for POP3 & IMAP) is built in; &, of course, it's a browser so you can get HTTP access to any web based e-accounts.
In some sense your original Q is flawed: e-mail clients don't do HTTP,
browsers do HTTP.
Most web based e-accounts are designed to keep you from using a mail client, it is probably a control thing -- they may want control of your e-mail, that is why you can't access & download it,
by design. If you don't like their policies, don't subscribe.
Instead of asking KMail to do webmail, use Konqueror.
Instead of asking Thunderbird to do webmail, use Firefox.
Besides, isn't there something strange about a GNU/Linux user even
having an MSN account, let alone accessing it from a Linux box?