In essence, if a structure can be found that cannot have developed through natural processes, wouldn't that force us to conclude that some non-natural, intelligent agent had to be behind it?
There are problems buried in the above. They are not new though.
1) How could we conclude that they cannot have developed through natural processes with out being in the position of giving an argument from ignorance? We know a lot more than we did 200 years ago but we still know far from everything about natural processes.
2) What do we mean by "non-natural"? Supernatural or just a natural intelligent agent? If it is a non-supernatural intelligent agent then it is acting through some selected natural processes. They would just be different than garden variety evolutionary processes. Even "artificial" selection acts through the usual evolutionary processes. We can't tell from the ongoing development of, say, dogs that it is not natural selection (can we?). We only know because we see the designer acting but the designer only selects from existing forms just as the environment does.
We might be able to tell if we were testing for genetic engineering though. Maybe that is what should be looked for.
If we did we'd recognize it in a number of ways:
1) a discontinuity in the genetics. This could take many forms but even some of them can be natural. I guess it isn't totally impossible for a virus or bacteria to carry a bit of fish DNA into ourselves for example.
More obvious would be the insertion of DNA that comes from no where else in any present or known past organisms. In other words a real discontinuity. Right?
2)A different basic mechanism such as a different genetic code altogether. Artificial life?