If you splice Everdream onto such a card, what happens? Do you draw the card from Everdream even if it leaves the stack first?
According to the rules for splice:
702.47e. The spell loses any splice changes once it leaves the stack for any reason.
(Perhaps this is meant to ensure that splice changes wouldn't apply the next time the spell is cast, for example.)
608.2k. If an instant spell, sorcery spell, or ability that can legally resolve leaves the stack once it starts to resolve, it will continue to resolve fully.