I thought the issue was more that we have no idea how to perform appropriate calculations for SM+QG? Maybe I'm wrong?
Each is “right” so long as they are consistent with our consensus driven glyph system that exists to normalize discussion, and should never be taken as a 1:1 model of reality.
The glyph systems are not a pipe.
There is not very good reason why the Higgs interaction is not called a force. I wouldn't read too much into it, tradition carries a lot of weight in how people call things.
We could even measure the attractive force it generates: jhttps://arxiv.org/abs/1601.05087
This potential takes basically the same form as the gravitational potential but with an additional exponential decay with distance due to the mass of the boson. Resulting in short-range interactions. So yes, in very short distances (~(inverse of the Higgs mass) ~ 10^-18 m) fermions do experience an attractive force between them due to the Higgs interaction.
It works in exactly the same manner the rest of interactions work, the boson (Higgs) is the mediator of the force.
The strong force is associaed with the gluon.
The Electromagnetism gets the photon.
The Weak force gets the Z and W+/- bosons.
Gravity gets the graviton, but that's mostly a placeholder since we don't know how to properly model gravitational effects with QFT.
Higgs doesn't get associated with a force (yet).