I’d say this critique by Elliott Sober is recommended reading on the design argument. It’s a comprehensive and detailed paper, but in good blog fashion I’ll try to briefly summarize the key idea here.
The design argument says the probability of our observing an object in the world (an eye, a bacterial flagellum) is greater given the hypothesis of it being designed as compared to the hypothesis that it was produced via a mindless chance process. The problem is that the argument itself lacks any support for this. What independent evidence do we have as to the designer’s motives and abilities? The answer is none, unless we bring in assumptions about the designer from outside the argument itself.
The argument appeals to an intuition or belief some people have about what the designer is like and what kind of things the designer would be capable of and motivated to create.
Interestingly to me, Sober turns around and shows how the same analysis can critique the argument against theism from the problem of evil (and/or the suffering of the innocent). For one to assume the existence of evil is an argument against the existence of God, one is assuming God has a certain set of abilities and motivations. What is the independent basis for these attributes?