yes it will work but you introduce an error in the cold junction circuit and the possible variable of oxidation of the rotary switch.
not sure about using SSR's as they are not a physical connector.
The question about 3.3/5 volt ........the Arduino board is a 5v device...the 4 channel breakout board is a 3.3 v device .....there is on the board the possibility
of making it 5v with a solder bridging circuit which must be configured { soldered } to make them compatible.
The sketch you provided compiles OK and as you've said works ...........no problem therefore ; its more likely a hardware issue , rather than a coding one.
Debugging the circuit is a lot easier if you have serial monitor routine to show up any fault conditions as the libraries provide.
I suggest you get Bruce to check the supply condition {3.3/5v} on the breakout board , and again as previously noted try bridging either or all of the TC 's
together without the water test .