I've read through this whole thread and much of what's here on both sides of the argument; I've heard before.
Both TOE and ID are based on "hypothesis" of origin.
TOE proponents want to argue that what appears to be "random sequence of DNA" proves their theory simply based on the innumerable possibilities of DNA sequence and given enough time those DNA sequences will "organize" into some sort of life form.
Yet the chance of this happening according to mathematical probability make this assumption impossible.
So we come to the conclusion that because the probability factors are not there mathematically; any DNA sequences has to fall within a formulaic perimeter. If all is based on chance than two cats breeding together could theoretically create an elephant. Yet, we never see that happen. All biological DNA sequences have perimeters for specific organisms.
Now on top of this; we can not demonstrably prove that "mutations" are beneficial to the organism. Matter of fact, more times than not, the opposite is proven. Yet changes in a DNA sequence that affect the appearance and function of organisms across generations can not automatically be assumed to be random. That is a product of the belief bias of the scientist and practically speaking; he who does not believe that particular bias is not going to be vocal about it, on account of the fact that he wants to keep his job. Regardless though; it is still demonstrable, that those changes still fall within specific parameters. Evidence being here that if they are too far outside of those parameters the organism does not survive.
Even just the simplest organization of any type of elemental structure (regardless of if your talking nature or man made mechanisms) inherently points to intelligent design, because random possibilities mathematically don't create organization of elemental structures, because if every time one "rolls the dice" is a billion to one probability of an organized structural outcome - it just never happens. And since we know that life is obviously elementally structured, it makes the TOE argument of "random chance" a moot point.
Now as to the OP's question of ID being presented as the alternative in the public square? I think you rather answered your own question on that. The inevitable conclusion that life is intelligently designed makes that life accountable to who or what ever designed it. Which also inevitably forces the definition of a Deity. And this is the real issue with ID being relegated to the margins of "science".
Now the TOE proponent argues that if ID were a "legitimate alternative" than industries such as medicine, agriculture, pharmaceutical etc. would "accept it".
Yet what that argument fails to see is the reality that all those industries depend on, is the predictability of the expected outcome. This is not foundational to TOE to begin with because the "predictability of the expected outcome" is anathema to a theory that's strictly based on chance to begin with. LOL. So strictly by default; in the realm of practical "predictability" all these industries demonstrate a reliance on the idea of intelligent design.