Why Run Supported Software
I agree with @canonizingironize, you are still going to be a target. There are so many people running out-of-date software that targeting them is lucrative. Also, let's not forget that you'll still be vulnerable to many of the newly discovered security issues. Many security holes are only found and fixed years after they were introduced. Hence, an adversary might target Firefox 58 but your Firefox 45 might be affected by the same issue. Also, I doubt anybody is going to throw away old exploits just because there are fewer people that are vulnerable.
I'm not sure what you mean by "Tor Browser is self-contained". Yes, Tor Browser inherits some sandboxing from Firefox, disables some features and adds some hardening. Sandboxing might well be broken since it depends on features provided by your unsupported OS. Furthermore, Tor Browser is based on Firefox and inherits a large number of it's security bugs. Just take a look at the published advisories and you'll notice that many of the issues apply to Tor Browser as well. Particularly, on lower security levels.
In conclusion, if you care about security, you'll have no choice but to update to a supported OS.
Answers, Question by Question:
What's the recommendation for older systems? Other than upgrading the OS to the latest rubbish.
There are no recommendation except DON'T! Everybody that cares even just a tiny bit about security must upgrade. In case you do not care about security at all, there is no reason to update your OS. Just as there is no reason to use Tor in the first place.
Keep the alpha build?
Unless you run a supported version of it, this is a terrible idea. Also, you should run a supported OS whether you're running Tor Browser or not.
Use the latest stable release?
Any other alternatives?
You could try a different OS.
And how can I turn off the flashing warning in Torbutton?
You answered this yourself already.