This is an interesting question. Although I am not deeply familiar with either of the games you mention, it seems that they are inherently time-based. There are also some random elements to CivClicker (e.g. traders that come through). Given these rules, I don't think a simple constraint solver would be sufficient to "solve" the game, but it might be a part of a more complicated solution.

However, there are lots of games that essentially only involve solving constraints. A simple example is Sudoku.