I think I already mentioned that I was worried about elemental damage, and I don't really have a good solution to it right now.
However, I do think there should be at least a few spells that need corpses for a couple of reasons:
1: Necromancy is meant to be about death and having any summons just pop up from the ground does somewhat ruin the effect. (Thematic elements can be important)
2: There are already other summoning spells (such as the summoner and possible spider spells) which just create the summon and having too many of them diminishes how unique each class is.
3: Having a handicap on a spell allows it to be more powerful than other spells. As such, spells that need corpses (possibly fresh ones) could have effects that are rather more powerful, again adding some impact.
4: This would establish corpses as a valuable resource that needs to be planned around, adding some elements of strategy unique to the necromancer.
5: Using corpses changes how the necromancers powercurve per battle in a rather neat way. Most classes are either okay at all points in a battle (fighters), good at the start but then becoming weaker (Sneaky classes and summoner), or have a peak in the battle(Pyro). (Note that is just from personal experience). Using corpses would make the necromancer start a battle much weaker, but grow increasingly more powerful, giving it a new powercurve that is strongest when finishing the battle.
Also, for some reason I'm a total sucker for that kind of necromancy, so I am rather biased. XD