You have mis-stated the Argument From Design. It is only weakly based on complexity, but instead is based on intentionality. If an object is most validly explained as a product of intentional design, then there is good reason from the existence of that object to infer a designer.
Complexity is only relevant in that it triggers a search for the understanding of the source of that complexity, because most things made by natural processes are not very complex. SOME objects we know are designed (baseball bat for example) ARE simple, and SOME natural objects are complex (see snowflakes) so simple/complex is only an indicator/trigger, not the actual condition that leads to a design inference.
The inference to design is a legitimate scientific reasoning process, and is a key feature of anthropology, criminology & forensic pathology and the SETI program.
Paley's Watch was a complex object, featuring aspects NOT found naturally at least in our experience (toothed gears, assembled movements, etc), but which ARE found in our intentionally constructed objects. Therefore, his inference that the watch was designed is entirely justified.
Life -- we don't have examples of it being designed, so the inference to a designer is far weaker. And -- there are so many oddities in life that don't fit with an intelligent design (viruses, poor organ/chemistry design elements, overpopulation/famine cycle for ecosystems etc), that design is -- suspect. When another mechanism was discovered -- natural selection -- which over sufficient time produces "designed like" features for any inherited property subject to variation and selection, then the inference to design for most features of life -- is no longer credible.
There remain three areas where an appeal to design may still be valid. The APPEARANCE of life has not been shown to be possible despite 3/4 of a century of abiogenesis research (see Freeman Dyson's The Origins of Life for a good discussion). The apparent purposefullness of evolution to create morality and reasoning implies design to Thomas Nagel (see Mind and Cosmos). And the Fine Tuning features of our universe (each of something like 30 parameters have to be in logically independent unlikely ranges for any kind of Life [not limited to organic carbon/water life] to be viable anywhere in a universe) make its intentionality for Life a plausible explanation.
The Fine Tuning argument is the strongest of these three, and has sparked a lot of speculation about natural mechanisms that could produce Apparent Fine Tuning. The current popularity of Multiverse hypotheses among cosmologists and lay people alike is because this provides a possible alternate method to arrive at Near Fine Tuning, by extending the timeline of a metaverse, generating variance, and then selecting from that variance. As an alternate explanation for Fine Tuning, it is logically possible. There has been a lot of push-back, however, from scientists and philosophers of science who point out that Multiverse plus selection is unsupported, and more importantly is UNTESTABLE (and therefore "not even wrong" as a science claim). Theist theologians can also point out that it is also an astonishingly complex hypothesis, which relies upon infinities. Complexity, untestability, and infinities used to be pointed out by non-theists as major weaknesses of theism.
Non-theists can point in each case to similar non-optimizations in these three areas as was seen for life. In Fine Tuning, our universe may be very unlikely, but it isn't anywhere close to well designed for life. In emergence of life the lateness and very local nature of that emergence suggests that life would be more of an afterthought than an original intention. And for morality and reasoning, the afterthought is even more pronounced, as reasoning took 4 more billion years to manifest, and is not present in most of life. There are therefore major weaknesses to an inference to design for each of these areas -- and the flaws in non-design explanations offered to date do not therefore lead to a justified default acceptance of a design explanation.
The possibility of our universe having been designed, to produce life, moral value, or intelligence, is therefore currently an open question.