What then might we conclude about the spiritual temper of much of the Christianity of the Middle Ages and early modern period, and its relative affinity with the spiritual temper of Indian Buddhism, from the ease with which Buddhist teachings could be absorbed without scandal, or even any sense of strangeness, by the Christian cultures of both East and West after the tenth century?
I don't think there's any great mystery here. The basic story is a striking one -- a wealthy prince is protected by his father from exposure to common human misery, but on coming into contact with sickness, poverty, and death, is touched by compassion so as to give up all his princely wealth, comfort, and power. Any religion with an ascetic component is able to make moral sense of such a story; from what I understand, for instance, there is a Shiite adaptation of the very same tale, analogous to (although not as popular as) the Josaphat versions.
But it's somewhat misleading to talk about how the story "could be absorbed without scandal, or even any sense of strangeness"; the story as we have it wasn't taken whole cloth from the original, but had already been adapted: Josaphat was Christianized in stages, losing distinctively Buddhist emphases and claims and as time went on retaining a connection to the original that was increasingly abstracted from anything especially Buddhist. Which brings us to the second question:
What then might we make of the delightful oddity that, in a sense, and admittedly under a foreign guise, the Buddha was for centuries venerated by Christians as one of their more beloved saints?
Any Buddhist reading the legend, I think, will find something of a shock at discovering that one of the most important elements of the original story is missing. The Buddha, on seeing human suffering, meditates and through his own efforts discovers the path to overcoming it. But one will search in vain for this in the Josaphat legend; not only does Josaphat not do this, it's clear from several things he says and does that he thinks it is impossible to overcome suffering through his own efforts. The right path is discovered not through meditation but through hearing Barlaam speak about the gospel of Christ. We are saved through faith, and faith comes by hearing: this is the message of the legend of Josaphat. Despite living the same general life, and despite the same themes of moderate asceticism, Josaphat is very unlike the Buddha, and to that extent the tale shows how the Buddha himself, however admirable, could never be venerated by Christians as a genuine saint. It puts in relief the fact that from the Christian perspective the Buddha's approach is necessarily incomplete, that from the Christian point of view Buddhist rejection of craving is good but not salvation. To make the Buddha a Christian requires making him a Josaphat.