Although ID claims to follow the evidence wherever it leads, much ID does not follow the evidence very well. ID is a self-described big tent, and includes quite a lot, so characterizing it is tricky.
Christianity and other theistic religions, as well as some other views, hold that everything is ultimately intelligently designed by God. The ID movement, however, claims that design has been implemented in certain ways and can be detected in certain ways, neither of which is required by the general belief that things are intelligently designed.
The proposed methods of design detection are problematic. ID promotes complexity and information as hallmarks of design. But if you think back to the last time you filled out tax forms or otherwise dealt with bureaucracy, complexity is no sign of intelligent design. In fact, design is more simple than randomness in important ways. To precisely describe the shape of a lamppost is not very complex mathematically. Describing the shape of a tree precisely, on the other hand, requires specifying where each branch is, where it bends, etc. All natural processes, including but not limited to mutation, constantly produce new information. The claim that an intelligent agent is needed to create new information is simply wishful thinking on the part of ID advocates.
In reality, design is detected in quite different ways from what the ID movement advocates. One way is through comparison with a large set of known designed and undesigned objects. I had a job doing design detection of this sort. It was an archaeological dig. As a geologist, I was acquainted with what rocks look like and was able to recognize rocks that looked like someone had chipped them to shape into a tool versus rocks without archaeological interest. Another way is to see if something specifically fits with a purpose that the putative designer would have. When ID claims to be simply following the data, it claims not to know who the designer might be. But that means we don’t know what the goals would be. ID assumes (along with some atheist claims to detect lack of design) that the design goal is like a human engineer’s. But that is not necessarily God’s goal. Empirically, it looks like creating myriad diversity of organisms may be a goal of creation; evolution by natural selection achieves that goal nicely, but is not as effective for hyperoptimization of certain functions - natural selection merely asks “does it work adequately?”
ID focuses on two different areas. One is front-loading / anthropic principle. Do the laws of nature point to a designer behind them? Many seemingly independent laws need to be extremely close to what they are for lifeforms similar to us to exist. Is that unlikely to happen in a universe that is undesigned? These are reasonable questions to raise, but the answers will be a matter of what seems plausible to each person. Science can’t get at these questions. We don’t have suitable data to calculate accurate probabilities.
The other ID focus is seeking for gaps within the history of the universe, especially within evolution. This somewhat clashes with the front loading - if the laws of nature were designed to enable our existence, they could be designed to bring about that existence through several steps without interference-style action. But the claimed gaps are not very good. There is also a wide range of views within ID from no gaps in evolution (e.g., current Denton) to gaps only in creating the first cells (Behe, Darwin in the 6th edition of the Origin of Species) to trying to attack all evolution (Wells, Evolution News).
In claiming to just be following the science, ID welcomes all sorts of beliefs, e.g. Raelians or Moonies. But ID also markets itself as Christian apologetics. These two are not compatible.