True, but to my ear at least, /eɪ/ is further from /ɛi/ than /aɪ/ is. Maybe more to the point, English speakers (American ones at least) are going to "realize" the name Dijkstra with an /aɪ/ dipthong. "Dayksra" would not occur to us to say.
Phonemes are a funny thing: if an American were to pronounce that last name perfectly (or a Dutch person speaking English, for that matter), other Americans would still hear /daɪkstɹə/, or at most, "/daɪkstɹə/ but it sounds a bit foreign".
Some sounds which we don't produce are audibly not-English-phonemes, like German ü or Scandanavian ø, but others are not. Another example of this is that native English speakers aren't usually sensitive to the difference between the r sound in our word "pear" and the r sound in Spanish "pero", but recognize that "perro" is a different rhotic sound.