Not sure why there is so much support for the dual-message, single siren system. One of the problems with having the same siren go off for both a severe thunderstorm warning and a tornado warning is that these two require different mitigation measures.
One means wake up everyone inside your house and get to the basement, or wherever is best, as quickly as possible.
The other means: "continue stay inside since you likely already were inside or made your way there recently because nature broadcast the tell-tale signs that a thunderstorm is imminent using a diverse array of messages decipherable to humans across all but perhaps one their sensory systems."
With any thunderstorm, the signs and dangers are obvious and the mitigation measure is clear. With a tornado, you need to seek unusually good shelter and the natural warning signs are not as obvious unless you can readily observe 360 degrees of the horizon from your location at a decent rate.
Save the sirens for the less common, less obvious, more dangerous, yet more mitigable threat of tornados.