The problem with your coin analogy is that any series of coin tosses is equally likely. The case of all heads is no more unlikely than any other string of results.
You are correct that all heads is no more unlikely than any other string of results. It's also true that the arrangement of letters in the works of Shakespeare are just as likely as any other combination of the same number of letters and it's it's possible that an infinite number of monkeys given an infinite amount of time could write the works of Shakespeare by randomly banging on typewriters. But the first thing that crosses your mind if you find a typed sonnet in the street is probably not, "I wonder what natural or random process produced this sonnet."
If you came upon a pile of exactly 1,000 nickels and all were heads up, underneath a large open bag of nickels tilted over on the shelf above the pile, would you assume that exactly 1,000 nickels fell out of that bag at random and all happened to fall heads up or would you assume that some person set up the pile by counting out exactly 1,000 nickels and placing them all face up in a pile with the tilted bag purposely placed there to mess with your mind?
That's the difference between possibility and probability. Yes, it's possible for exactly 1,000 nickels to fall out of a large bag of nickels into a pile and for every nickle in that pile to fall face up. But it's also very improbable. When faced with something that's possible but improbable, you can attribute it to chance or design, and which one you err toward will be based on secondary evidence and other assumptions. For example, if you came upon that scene in a locked bank vault, knew that the bag of nickels was there the night before and the pile was there the first thing in the morning, and had video camera evidence that nobody entered the vault, it would make sense to assume it was a random event. If, however, the bag and pile are found in a room in a house inhabited by practical jokers, you'd probably assume a person set it up. And in between, there is a lot of uncertainty.
ID people look at the probability of random mutations and natural selection producing a certain outcome to be unlikely enough to warrant looking for another cause. If you already assume that no other agent could possibly be in play, then it's easy to accept that any possibility as proof that it did happen, just as finding a pile of coins in a locked vault with no other explanation would make a random explanation seem quite plausible. If, however, you don't presume that the universe is godless or that random mutation and natural selection are the only means that exist to explain a certain outcome, then it's not unreasonable to explore other explanations, much as a person who found 1,000 face up nickels might look for a human explanation. It's possible that it happened randomly. But without additional evidence, does mere possibility warrant any certainty that it did happen or does the implausibility of the result warrant some investigation for some other cause?