This argument still falls apart when faced with simple stratigraphy. You find corals where you would have found warm, sub tropical environments. In the same place, you tend to find limestones, not mudstones/sandstones, and Geochemical data in Oxygen Isotopes can be used to track the temperature, so we know it was warm.
When we find coral fragments halfway up metamorphic mountain range, you might have a case...
Your argument still doesn't explain entire reefs in mountainous regions. I've been up a big old hill (not tall enough to be a mountain) in Spain to find a huge reef system, and as you walk up the side, you can track the growth of the reef as sea level changes.