Actually, the question should be "why do other dairy products spoil so fast?"
If I remember correctly, the spoilage of milk is caused primarily by lactose, which breaks down into lactic acid over time due to the presence of Lactobacillus bacteria, which thrive in that environment.
Butter is mostly fat, and fat does not go bad (it does eventually, but not in the same way or at the same rate - see below) - lactose and sugars together, on the other hand, go bad quite fast. That is also why cream lasts longer than whole milk, and whole milk lasts longer than skim milk. It is also the reason why lactose-free milk seems to last forever - there's nothing to break down, nothing for the bacteria to feed on.
When butter does go bad, it's generally not even for the same reason - it's simply due to oxidation which causes rancidity, similar to a typical vegetable oil.
In short, butter lasts so long with or without refrigeration because proportionally speaking, it doesn't have as much of the lactose that causes other dairy products to go bad.
With the recipe you mentioned there is no indication as to what temperature the cream cheese and butter should be at and I'm guessing that's one possible item making the difference. Since it looks a little like a creaming or foaming (sponge) method for whipping ingredients, I'd try to make sure my butter, eggs, and cream cheese, but especially butter, were around 70 degrees F. You want them to be able to incorporate air. The first link talks about how to get things crispier or fluffier based on how long you cream, etc. Then you will want to play around with batter temperature when it hits the oven.
Also note that more fat in a cookie recipe, at least, is equivalent to crispier. Are you subbing nuefchatel cheese or low fat or no fat cream cheese for the cream cheese? If so, don't. Moisture in sugar has an impact so make sure you aren't subbing any other sugar for powdered.
Real gooey butter cake is a yeast cake rather than the rip-off versions that use cake mix and not really all that much more complicated, although more time consuming. You may want to try doing a more authentic recipe, or at least one that uses a yeast cake. I have had success with the one linked.
Best Answer
I have dealt with the same situation - very warm in the kitchen and the butter gets super soft, sometimes with a very slight bit of melting. This has never negatively affected the outcome of my cake.
But, anytime you think it may be too warm in your kitchen, you can put a mixing bowl in the fridge or freezer and let it get cold while the butter softens. Then, when ready to start creaming the butter and sugar together, transfer the butter to the cold bowl. This ensures that the butter will not continue to get warmer during the creaming process.