I just want out of the MS enclave! I like W8 - have been using it on phone for a couple of years, but I don't like the way all the apps seem to be tied up in that way - it was partly to avoid that kind of stuff that I kept clear of iOS. There is a still a Yahoo Mail app (2015) in the Store and I've read somewhere that once you've installed an app and attached an email account, using your MS account, you can disconnect to a local account, and still access email, but I can't find out how to do it.
Ed
I don't understand why you feel like your trapped in ecosystem. In fact, Microsoft is just following Industry where Google you need a Google account, Apple you need Apple ID, BlackBerry you need a BlackBerry ID.
One good thing is unlike the others, Your Microsoft account does not tie you to particular operating system or phone.
I don't know what to say but in order for you to fully use Windows 8 and beyond you need to use Microsoft Account. You don't need to Microsoft Account services. As you stated you need to login to get your apps that you previously installed that were removed from the app store.
You may be right that you can install the Yahoo mail app but generally when a App gets pull, the app will soon have its API shut down after some time so your best bet is to use another mail client.
Also your making a big deal out of nothing, if you don't want to sign into Microsoft account. You have the option to use only desktop apps.
Is your tablet allow desktop installs? There might of been a workaround but probably after some time, it got patched so you are unable to do that. On top of that, you have to wonder about the future update to Windows 10 will be more tied to MS account or Azure ID for work.
Best mainstream Alternative mail clients is either
Microsoft Outlook 2013 or
Mozilla Thunderbird.
I don't recommend you installing Windows Live mail client since it is not getting much updates. The future plan for Microsoft is to bring Outlook as a Windows Store App in the Windows 10.
I also have yahoo account as my main account but I am having struggle in moving everything to my Outlook.com account. so I use both Outlook.com and yahoo.com and pay for $20 each for ad-free experience. I personally use Outlook 2013 x64 and I use Lumia 925 with Windows Phone 8.1 Denim.