What I've done is send email to Gmail and force it to use the basic HTML-only website (there's a trick to forcing it in the browser by disabling Javascript, loading Gmail to get the error, and then saving the preference for basic HTML view, assuming Google didn't change things again). There are, additionally, other standalone email clients that can be configured not to download anything beyond text (and probably filter attachments too).
An idea that crossed my mind was that a text-browser or email client would certainly be an option. I didn't mention it because text-only clients are likely familiar to just a small subset of computer users since in today's world most people have probably only used GUI. Sure, there's emacs, with several email options--or mutt, pine, etc, but few would go there. I really don't want to either.
If not using Gmail, I don't think disabling javascript would work--most web sites seem to depend on it for rendering the page. You could be right that there are GUI email clients that could read just the text/html parts of a message without attachments. I don't know what programs are available, which do you know of?
With all the variations in POP3, IMAP, SMTP settings used by ISP email servers, getting the client program configured takes some knowledge and care. But having done this before on various platforms, I'd say it's not really terribly complicated and might be a better way to go.
It may be a different thing vs. web-mail, in that web-mail leaves all the files on the server. The browser allows examining the files without downloading them, whereas the email clients I'm thinking about download the files to the user's storage. Perhaps that's configurable, anyway, if it matters, it's something to be sure about.