The easiest way to test the whole thing is as you have done, bridge the two connectors that gon on the switch if it then works the fans the switch, is either not up to temp or duff.
For an accurate test drop in a saucepan of water na heat up with rad switch with wirea ttached to it's terminals and a multi metre, on a resistance setting, you will soon see it swing/buzz whne then contacts close as they do when up to temperature.
Are you sure you are letting it get hot enough, whne my sapphy is on the "N" it is only at 93C halfway through the N.
tabetha