I wish I could help since your post was incredibly useful to me (described below). I extensively use Kindle's "experimental browser" and have had problems with it, but not the ones you describe.
I don't think this will help with the cert issue, but I always keep javascript and images off. Some sites with JS enabled crash the browser. A few sites, Gmail being one iirc, require enabling JS temporarily. When I used Gmail, I was able to access it on my Basic Kindle, v. 5.6.5. I never have had a problem with Twitter or Wikipedia. My main problem with certs is the captive portal, or whatever it's called, of some hotspots. Sometimes I get the option to proceed past the portal if I accept the danger; other times I can't get past the warning of a bad certificate.
You mentioned mowser.com in your original post. I'd never heard of it, but this is a great service. I am always looking for ways to make webpages display nothing but text. For that I use "article mode" or "reader mode" in graphical browsers and still use text browsers like Lynx.
Mowser works beautifully even on my desktop PC to display webpages the way I like them to look. It's also handy for my mobile devices. I have already bookmarked many of my frequently visited sites prefixing their URLS with
http://mowser.com/web.php?raw=true& . So thank you for that site. Not to be greedy but are there other sites that do the same thing as mowser?