Does crustal thickness have anything to do with how life existed and sustained on Earth?
If the crust were the thickest layer or Earth, several things would happen:


It wouldn't be a "crust" any more, by definition. Because this is what a "crust" is: a thin layer on the exterior of something. However, if we assume that the mechanical properties of the crust (being cold and brittle etc) would extend deeper in the Earth, the following applies.
No mantle convection, or at least mantle convection weak enough to probably not affect the surface. Therefore, no volcanoes, no mountain building, no subduction, no recycling of volatile elements, no sub-seafloor hydrothermal vents.
If it's cold enough, the core probably solidified and there is no magnetic field.


A good example would be Mars. A planet hypothesised to have tectonic activity in the past, but not any more. The crust of Mars isn't the thickest layer (again - think of definitions), but it is thicker in absolute and relative terms when compared to Earth. I will leave the implications of "Marsifying" Earth on organisms for you to figure out.