Goldfish can actually live in temps even higher than almost any tropical fish can tolerate. Small ponds baking in the sun in southern Florida get hot. And yet the goldfish in them are loving it, even breeding. When the temps get that high it is not the temp that kills goldfish, it is the lack of oxygen. Keep the aeration up and there is not an issue. Fishbase.org (one of the best resources for ACTUAL max sizes, natural conditions, etc.) lists goldfish as going up to 100F. This would explain their ability to be fine in well-aerated small ponds in hot climates. Anyone saying they can't tolerate temps above 70F is improperly informed. They have no issues at all being in the upper 70s and even well into the 80s.
Diet is not as much of an issue as many make it out to be. I only feed New Life Spectrum to all my fish. The actual formula effectively does not vary from one variety to another. So my fish are all getting the same diet (and doing better on this high quality diet than any other diet I have ever tried).
Goldfish are not dirty. A high quality food that they use properly (as in it does not just come out the other end and offer little for them to actually use) cuts down on their waste a lot, just like with any fish. They do not produce any more waste per weight than a tropical fish. Bring cichlids into the conversation and the goldfish may be even cleaner.
Behavior is one of the biggest reasons. Long-bodied goldfish are aggressive eaters. Stick them in with some laid back tropicals and the tropicals could starve, but most tropicals would be fine. Plecos will suck on round-bodied goldfish doing severe damage, even in just one night. Many fish are too nippy and will peck at the slime coat of the goldfish, especially the slower round-bodied breeds.
In reality it is possible to safely mix goldfish and carefully selected tropical fish. In addition, there are a number of fish that naturally prefer cooler waters that do very well with goldfish: guppies, platies, mollies, swordtails, danios, white cloud mountain minnows (all actually not tropical but sub-tropical and prefer temps of about room temp just like the goldfish), dojo/weather loaches, and rosey reds. Some of these may be too fast and nippy to be in with the slow round-bodied breeds, but are still compatible with goldfish in general.