Can problems on the secondary side of the transformer cause it to cook and if so, what sort of things?  My first thoughts were that there is an issue on the primary side of the transformer.  It is the bottom section of the transformer that is cooking.

In 99.999999999999999999% of cases a transformer (TX) will fry the PRIMARY (mains connected side) if there is an excessive load on the SECONDARY - this is VERY normal.

As this TX is basically directly connected to bridge rectifiers that would be my first port of call unless there is also some sort of surge device (MOV etc) ACROSS the secondary winding? There really is very little else that will cause a catastrophic failure such as this.

The mains fuse IS NOT THERE to save the transformer OR even your game. Its ONLY purpose is to prevent your house from burning down - do NOT ever think the mains fuse will save or protect your game - it won't and isn't designed to - that's why I suggested putting a much lower value mains fuse in with the secondary DISCONNECTED as a test. The lower value is very likely to pop if there is a fault in the TX BUT also it will likely stay intact if the TX is OK but with NO LOAD at all.