Bit of an update: I installed SetEdit (Settings Database Editor) and confirmed that there doesn't appear to be a "always_on_vpn_app" setting in the Secure section there either.
But here's where it gets weird. If I use tasker to READ the setting into a variable and I enable Use Root, I get the current value. IE if I set up OpenVPN Always-On through the regular Android settings, then get the setting, I correctly get "net.openvpn.openvpn". If I turn off Always-On and get the value its "null". Note that this only works with Use Root, if I try without, I just get blank as the value. But I have tried setting the value using Use Root, and there's no error, but it also doesn't apply the change. If I go into the Android settings, it's still turned off. If I retrieve the value though, it will be what I set it to (until I change it in Android Settings again).
So the value is there, but it is also somehow not there. And it works, but yet it also doesn't. Go figure. Anyway, wanted to add that additional information.
One last thing, I can set "always_on_vpn_lockdown" just fine. That works perfectly. And it shows up in SetEdit too.
Thanks in advance for any help, I'd really like to get this working!