Of course the idea is an analogy, and I strongly hold that all analogies are wrong when it comes down to it. They can be useful to teach real learning, but they can also be used to propagate nonsense in a plausible way. But setting that aside here is a question.
When Paley found his mythical watch, was it still ticking?
I think it would make a difference. After all, the same analogy applies to the world around us in some ways.
If his watch was ticking, then it might imply that it had a designer, but it would also imply that it had a winder. Within the previous day or so, somebody must have intervened in the existence of the watch and wound it up. Without that intervention the watch would have stopped.
If his watch was no longer ticking then the conclusion about the designer is unchanged (and incidentally it is still wrong, being only an analogy). But it would suggest that the designer had gone away and left his invention to fend for itself.
This is the type of world that we live in. The designer might have made the world - although there is not a jot of evidence to support this idea. He might seem still to be involved, but actually the designer no longer gets involved in it. We all know that the studies of the effectiveness of intercessory prayer have demonstrated no value, or even a slightly negative correlation in the outcome of those prayers. (See here for an example that shows no significant effect.)
God doesn't answer prayers, but the people who are praying still find ways to claim, without convincing evidence other than 'testimony', that it has made a difference to their lives.
Isn't the mind a wonderful thing? It can make up credible explanations for uncaused non-events.