In Preludes and Nocturnes, in which Dream is captured, the reason for the mistake is not described. However, later in the Sandman series, during a flashback, Dream recalls his imprisonment, and hints that it occurred after a significant event, far from Earth, which he had attended. This event had weakened him considerably, and he was returning in a vulnerable condition. It was only because of this temporary vulnerability that the spell cast by Burgess was able to capture him, or one of the Endless at all. I am seeking more specifics on the event in question, which I seem to recall is actually disclosed later in the series, and will post it once found.
The question of intent is a very interesting one. On the surface, Dream's capture was a simple accident, one he had to spend quite some time and effort in rectifying. However, it was only because of his imprisonment that
his successor, Daniel, was brought into existence, by being gestated by Lyta Hall in the pocket dreamworld controlled by Brute and Glob.
As you had noticed, many events that seem random end up being used by Dream to his own ends. At the end of the series, when
Dream chooses to end his own existence by allowing the Furies to bring about his death, he seems aware that the child Daniel will take his place, as indicated by his gifting of the Eagle stone to Daniel before he departs. If Daniel did not exist, the duty-bound Dream may not have chosen to lay down his burden in the absence of a successor to pick it up.
Thus, the end of the series was dependent upon the beginning, an apparent accident and misfortune leading to the most powerful and pivotal event in the existence of the main character. I consider this both a highlighting of Dream's foresight, and the writing ability of Neil Gaiman.