You are right. I didn't think of anything to fix this during the jam but now you said this the solution is clear. 

I will detect if there are no possible circuits and if so tell you then scramble all the pieces around. I think one of the Bejeweled games did something like this. I'll make a new version tomorrow as a separate project.

When I do this I'll see how common it is that there is no path. There can sometimes be difficult to spot paths that go around the whole board.

I implemented swapping the board if there are no possible circuits and I could not get it to trigger naturally, even generating 100s of thousands of random boards. So I am quite confident that there likely was a solvable path.