I ended up giving up and installing Safari 9.1.3 1.5 years ago, although it's showing it's age more and more. The sites built using more sophisticated kits, rendering models load incorrectly. However, it still works on simpler sites and there's an ocean of those. Using with Mavericks it's not just Safari 9 but WebKit engine that powers other applications (iTunes, App Store), and also TSL protocols that are hampering the browsing more than anything.
However, it seems there's a light at the end of the tunnel: the user
wowfunhappy, the guy behind a resurrected Weather Widget, has compiled and built a proxy-server based on open source that maps connectivity of macOS 10.9 and older into the newest protocol of TSL 1.3 which itself hasn't yet become the commodity with web-developers, so it's a giant breakthrough for those who either can't or not willing to update. Look at
Squid - a web-proxy and follow the instructions strictly.
You also can put your own domains in Squid's configuration file
squid.config that are excluded from proxying: that way, and experimenting on my own, I was able to keep iMessage, Facetime, MAS useable but also resurrect iTunes Store and Genius in iTunes 11.4 (purchasing and previewing audio-tracks still isn't possible, though), but most importantly
Safari wasn't able to establish secure connection messages are gone for good: only the modern design is an obstacle (that is, you still can't play YouTube videos, and use new Reddit, many commercial, e-banking decks, and some super-fancy sites). I installed the proxy on my Lion machine too and was able to bring back iTunes Genius there. Safari 5 even can open some sites it failed on before. I use Firefox as my default on both, though.
Both Safari 9 and 5 now pass SSL compatibility checks successfully - all tests show up as green (for example,
howsmyssl verifier)
Drop me a PM if you need it.