Is eleagnus wrong or just simplified?

Simplified past the point where it is still the name. The ae is a digraph carried over from Latin, which took it from the Greek ai of elaia, olive. In English botanical use the pair says a single ee, so writing one e for one sound is a reasonable thing to do and it is the commonest thing writers do to this word. It is also the one change the name cannot survive. The code that governs botanical names lets you write the old printed ligature as two letters, ae, and it does not let you drop one of them, so Elaeagnus is spelled with four vowel letters in the first five characters whether or not you can hear them. The same digraph turns up all over the Latin plant vocabulary, in Aesculus for the horse chestnut and in Caesalpinia, and it behaves identically each time: two letters, one sound, no warning.