Why doesn't EDTA have a double bond?
A lot of non-systematic chemical names don't describe the composition of the compound, they describe what the compound is synthesised from. That is the case here since EDTA is derived from ethylene by adding two units of the diamine.

It isn't that uncommon in chemical names. Polyethylene, for example, is (simplifying only slightly) a long chain of carbons connected by single bonds. But it is made by polymerising ethylene. Hence has a name that doesn't describe what its structure is but where it came from.