TSE (Ethernet) - Core does not transmit data on RGMII TX, rest works perfectly fine

Since it works in simulation and not in hardware, maybe double check that it truly works by using the (post-synthesis) Gate Level simulation. If that doesn't work, it points at a difference between your testbench and synthesis which you ought to be able to resolve. If it does work, it points squarely at the physical hardware differences which maybe will need more investigation.