Geist could work, if you were willing to go with a slightly less technical feel. Geists get their power through their bonding, the power is inside of them - they need Keys to access the power, and if memory serves you do need a physical object, at least at first, to be able to do that, and Geist is definitely more lighthearted on the NWoD side, but you are still a supernatural creature, not a human with shiny objects.
Upon further review, I've found a few more possible entries, along with some that would need...work.
BESM - it's definitely cartoon-based; better for The Real Ghostbusters rather than the movies, but 3rd edition would let you build that sort of character.
Cypher - this system has had two major incarnations (Numenera and The Strange), but I know the system itself is out there, floating around, after the creators released the previously-mentioned settings for it.
Mutants and Masterminds - you were already talking about building a world anyways, so the fact that it's slightly more calibrated for the superhero genre shouldn't be too much, right?
PDQ - I've got the base book floating around here somewhere, but the major book I usually keep in front of me is the Ninja Burger guide, which is one of only a few settings I've seen PDQ used for.
And the...ones that need work.
Blades in the Dark - it's meant more as a nod to games like Dishonored and Thief, except with a crew of smugglers/thieves/whatevers, but there is a significant supernatural presence in the game, and players themselves don't have the power to deal with them, they need special equipment for that.
FUZION - I used this a long, long time ago...unfortunately, the only FUZION book I have left nowadays is Bubblegum Crisis, and mecha-cyberpunk is not precisely the genre we're going for here, but the basics are in there.
RWBY - the thing of it is that the game book (the one that exists) is pretty tightly tied to portraying a show of the genre that RWBY is...which is not precisely conducive to the genre that I picture Ghostbusters to be, nor the action level.