Great question. Looking forward to hearing thoughts from others.
Most of ID leaders are OEC, like @vjtorley and @Agauger are inclined. They genuinely have had legitimate and valid concerns about atheist evolution, and feel that EC has often given up too much. They also have theology is in many ways incompatible with EC. For example, even though a historical Adam is technically within BioLogos, ID leaders (in my experience) have generally felt that a historical Adam (ancestor of us all, specially created) is required. There are exceptions (e.g. Behe), but that seems to be the pattern.
At the same time, a very large portion of the ID base is YEC. They do not score any points for arguing for an old earth. The whole rational for forming ID in the first place was avoiding internal debates about the age of the earth in order to focus on fighting more important battles, like opposing atheistic evolution. To his credit, Philip Johnson made an effective case for including theistic evolutionists like Behe too.
For that history, ID as social movement works because it is triangulating. They argument to YEC’s is that they should put aside their disagreements with OECs (and even some TEs), in order to made a more effective case against Dawkins and evolution.
Where it gets interesting is EC, Francis Collins, and BioLogos. TE in this form is a creation of the ID movement; it arises historically as a reaction against ID, even more than YEC. BioLogos, for example, has really be defined by opposing the excesses and errors of the ID movement. They would not exist however, if not for the ID movement in many ways. It was because of the Dover Trial that the Language of God was published in the first place.
So EC has honestly been attacking ID for far longer than ID has been attacking EC. From talking to people like John West and David Klinghoffer, I think they’ve legitimately seen themselves as defending themselves from EC scholars. They have struggled to understand why Christians would oppose their efforts in making the ID case. It is a big change, in truth, that the new Crossway TE book was published by ID thinkers. This is really their first concerted effort to engage with EC theology in a public way.
That is a good thing. Getting the theological differences on the table will be how we sort through them. In the past they have resisted engaging theology, even though that is really where the crux of the disagreement is. Perhaps @Agauger can comment, but perhaps they felt they need to engage directly with BioLogos in these terms. It certainly served their social strategy of triangulating between evolution and YEC to negotiate internal truce between YEC and OEC. It is just now that they are clarifying that opposing EC is just as important as opposing atheistic evolution; and that opposing EC will require a theological response too.