I don't use either Ring or SimpliSafe but my WAG is that they won't work together.
I moved over this last winter and went thru something similar to what you are facing. For many years I used SmartThings (tried Wink once for a few months but went back to ST) and had a fairly extensive setup with many lights, sensors, locks and cameras. When we moved I made the decision to scrap it all and start over. Like you are headed for, my previous setup had a mixture of products from a variety of vendors and I was constantly having issues with things that had been working suddenly stopping. Sometime I could get them working again and sometimes not. Almost always it was a case of one vendor making a change in his software/firmware/driver that wasn't compatible with the other components in the system.
When we moved I threw out the SmartThings hub and settled on an Amazon Echo Plus with Zigbee. Zigbee is a mesh network that is self repairing and every new device you add extends the reach of the system. At that point Amazon had just implemented their sensors API and I knew that it would take a while before I would be able to duplicate much of the functionality I had prior to moving, but I was/am willing to wait. My rationale was that Echo/Alexa is such a huge installed base that other manufacturers will be forced to adopt the Zigbee interconnection system over Z-Wave or others. We've now reached the point where I'm starting to see that bearing fruit. KwikSet and Schlage have introduced Zigbee based locksets and many other device categories are starting to become available.
On the downside, Amazon isn't upgrading things on their side as fast as I'd like. Things like adding support in routines for device triggers and conditional triggers aren't available yet but I feel sure that it's just a matter of being patient.