jagman,
For whatever reason, it seems that when one company starts doing something, they all do — even to the point that they will sometimes abandon what they
were doing in order to concentrate on their 'new' thing. Hence, Gmail, Yahoo, AOL and Outlook all introducing their new interfaces and ultimately dropping their older ones that most users
didn't want to lose (as in your case). (Gmail does still have a "basic html" option, I believe, but it may not have what you're looking for.) Result: it may be hard to find a free webmail client that has many of the functions viewable on its interface because the above-mentioned providers are too busy competing with one another than listening to their users.
There are some webmail interfaces (like OpenWebMail, SquirrelMail, Horde, Group Office, Roundcube) that may have the kind of visible functions you want, though I don't know how good they are at showing 'conversations.' And I think they all allow turning off 'auto-save.' Since Vivaldi is free, there's no harm in signing up for an account there and test-driving its Roundcube interface to see if it has what you're looking for.
As for the other interfaces, I only know of them being available at premium email services. E.g.
Eumx.net ($16/yr) has Squirrelmail, Group Office, Roundcube, Horde and OpenWebMail; PolarisMail has Squirrelmail, Roundcube, Horde and AtMail.
PolarisMail's basic account is $1/month with excellent customer support and a choice of several interfaces or using your own email client!
VFemail has a very small free level (perhaps you could sign up for this for testing purposes) and similarly inexpensive account levels. Also, FastMail and Runbox have account levels below $20/yr.
Many people balk at the idea pf paying for email. But consider that you can have a
year's worth of service — including
personal customer support! — for less than the cost of ONE home-delivered pizza. Depends on how important email is to someone, I guess. Just something you might want to think about.
Also, I don't know if I'm misunderstanding something you said — and I'm not sure how much you know about these things (you may know a lot more than I!) — but just to clarify: Using an offline client won't mean that you won't have to have any emails on the provider's servers. In fact, by setting up an IMAP account in a downloadable email client, you will have your email 'mirrored' in
both locations: on the servers and on your computer. If anything were to happen to your computer - or if you run into issues with the offline email client itself that you can't resolve - you can always just download another client and set up the IMAP account and it will sync with the webmail servers and download the mail that's still sitting on them. And the advantage of the offline client in your case is, I believe, that most of them have more of those visible functions that you're looking for. And they're usually more customizable along those lines, esp. Thunderbird.
One last item: I see you mention Gmail and Yahoo giving you 'protection' and your not having to back up emails. I take it you mean you feel you would rather trust an email provider's own servers to store email than trusting your computer to do so. I feel the same way. Just be careful, though. We all know that using any email provider
exclusively is risky. Provider problems can cause the loss of email on occasion, even with their back-ups. And if an account actually gets hacked the damage may be irreparable (emails and files deleted permanently, etc.). It's always wise to back up one's email
somewhere, either using an email client (with the option of setting up a POP account, if it's only being used for offline back-up) or auto-forwarding to another email service, or both. That way all your stuff is in more than one place. Better safe than sorry.