Perhaps. We don't know, yet.
everything is a mixture of massless things
Masslessness is more elegant, more mathematically pure, so I would expect truly fundamental particles to be massless, yes. But that's OK; after all, photons are massless.
and the Higgs boson which is also rather elusive?
The Higgs boson is also a by-product of electroweak symmetry breaking.
Well, then that just puts the icing on the cake. Something that no longer exists came from nothing that did not exist giving us something we haven't found.