Unless you count Hotmail, I've not been using Web based mail readers for 20 years like OP, because in the interim I moved them all to the Mail app (around 2005). I have about 7 Gmail accounts (G Suite for Education, G Suite - for Business, G Suite - for non-profits, Google Apps - free grandfathered from long ago, 2 different public Gmail accounts - due to different activities and one general purpose Gmail account), one iCloud account (from the .mac days), now the G Suite ones work best with selecting "Exchange" as the type of account. I also have my original Hotmail account and two Outlook.com accounts (needed that for two subscriptions of Office 365 Home - we have 8 PCs/Macs so this is actually really good for us). Everything is in the Mail apps of my Mac, iPad and iPhone. Sometimes 2-Step Verification gets in the way of the Outlook.com ones since there is no easy way to enter in the OTP (one time password - or authentication code). It takes a lot of tries to get that right. The biggest advantage is that sometimes I get community service emails into my iCloud email but one of the public Gmail accounts is just for that. In the Mail app, it's real easy to move from the iCloud account to a folder (ok, technically "label") in my community service Gmail account, by just dragging and dropping. Additionally, even if within (outside works as above), the Mail app in MacOS and iOS "remembers" where you've filed some email (it looks like 3 saves causes it to offer up the last save). Right click oh the mail and if you want to file, in the Mac app, there is a choice "move to <mailbox name>" and on iOS, it comes up in the file to folder option.
For Web based access, iCloud.com is usually what I use. It won't have all the different accounts available, like the Gmail, Hotmail or Outlook.com. Gmail (G Suite and Public) and Outlook.com have a few more features in the Web version (like out of Office notices, auto forward, etc.) Rather, both the Mail app and the Web versions are complimentary. I'd like to offer up that 2-Step Verification or better yet, 2-Factor Authentication when available (I even use it on Evernote, Amazon, cloud based storage, any where possible, I've enabled it. Now I know that 2SV and 2FA are not end-all-be-all totally secure, but at least in the basic sense, you'll at least be aware that someone is attempting to access your account and have your password, where you can change it before they do.