No
You must choose a specific skill to buff, so it doesn't apply to Initiative ability checks (which are Dexterity checks) like in the 2014 version of the spellcantrip.
Another difference between editions is that it now works on all checks with that skill for a full minute.
So your DM is right.
Side note
Technically speaking, there's no skill checks in the rules as a term. In D&D 5e, all checks are ability checks. This is why every check in published materials is listed as Ability (Skill), such as Dexterity (Acrobatics).