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?
But this is were ID parts company with reality. Selection is the very things that shapes the outcome. You can't formulate any mathematical rule that distinguishes a produc of selection from a product of some other cause. the biggest problem for ID is that we can see selection at work in real time. ID does not bring any ongoing process to the table.
The problem with this line of argument is how do you calculate the probabilty of mutation and natural selection forming a given biological system? In the case of the coins, it's easy to calculate the probabilty of a given sequence of coins. However, natural selection provides a non-random input to biological systems. In your analogy, this would be akin to replacing some number of nickels with two-headed ones. If we replaced, for example, 950 of the 1000 nickels with two headed ones, is it still more reasonable to conclude that the pile is a result of design? How about replacing 990 of them? At what point do you draw the line?
However, the real problem with your whole argument is really that you consider some equally probable events to be more improbable than others. Which events are considered more improbable? Namely, those that result in a recognizable pattern. Humans are good at pattern recognition. The problem is that we are so good at it that we recognize patterns where none really exist. Just for another example, if you play the Powerball lottery, I'll bet you've never played a ticket with the numbers 1,2,3,4,5 and a powerball of 6. If not, why not? That ticket is equally unlikely to win as any other. Just intuitively, we consider such recognizable patterns as unlikely to happen randomly. However, given enough drawings of the Powerball, it becomes likely that this set of numbers will indeed be drawn.
Similarly, I once had a statistics professor assign a class the task of flipping a coin 10000 times and recording the results. Inevitably, some people didn't actually flip the coin and just made up results. Just as inevitably, the prof could always tell who faked it and who really did it. How could he do this? In a series of 10000 coin flips it's almost inevitable that there should be some sequence of 6 or 7 heads in a row. It's also human nature that if you were trying to fake such a random sequence, you would NOT put 6 or 7 heads in a row in the sequence because you would consider this unlikely to happen.
The point is that arguments based on improbabilty are weak ones. Very frequently, we have no real basis for calculating the probability in the first place. Even if we do, we have little basis for saying that a particular configuration for a biological system is any more unlikely to have happened by stochiastic processes than any other configuration for that system. If that's the best that ID can do, then biology will undoubtedly never take it seriously. It's akin to stating that the Powerball drawing must be fixed because the numbers drawn were 1,2,3,4 and 5 with a Powerball of 6, and this is exceptionally unlikely to have happened without fixing the drawing.