Subfloor and Batten Construction of 1950s Hardwood Floor
Your situation is not typical unless those are intended as sleepers (used to correct out of level joists) or furring (used to raise the plane of a surface in order to flush with the plane of adjacent finishes). In the first case, you would still be missing a layer of sheathing above the sleepers as sub floor. In the second case, underlayment should have been full coverage. As you are discovering, what they've done leads to issues for a typical lumber flooring plank spanning across a typical joist layout.
Some quick points:

It would be impractical to fix this in any other way than to live with it or pull it up  and relay your floor properly.
The brown paper is probably kraft paper for a little moisture resistance from the basement humidity. Most likely, whoever installed this was cutting costs and willing to live with the consequences. Without sub floor to support the paper, it's a miracle they could lay it at all.
There is no reason to run electrical between flooring and subfloor.


Since your Plan A didn't work, maybe this could be your plan B:
You might try finding any nails that are sitting on the side of a joist, which squeak from rubbing under normal circumstances as you walk across the floor, and any nails that are sticking out from the side of the joist, which squeak under normal circumstances from the inadequate withdrawal resistance. You can try pulling them through with a pair of end cutters.
Other than that, try adding a picture to your post and maybe it will allow for a solution as the description you've given is currently explained.