Understanding what makes Bi$_2$Sr$_2$Ca$_1$Cu$_2$O$_x$ (Bi-2212) the only high critical current density ($J_c$), high temperature superconductor (HTS) capable of being made as a round wire (RW) is important intellectually because high $J_c$ RW Bi-2212 breaks the paradigm that forces biaxially textured REBCO and uniaxially textured (Bi,Pb)$_2$Sr$_2$Ca$_2$Cu$_3$O$_x$ (Bi-2223) into tape geometries that reproduce the strong anisotropy of the native crystal structure and force expensive fabrication routes to ensure the best possible texture with minimum density of high angle grain boundaries. The biaxial growth texture of Bi-2212 developed during a partial melt heat treatment should favor high $J_c$, even though its $sim$15$^{circ}$ full width at half maximum (FWHM) grain-to-grain misorientation is well beyond the commonly accepted strong-coupling range. Its ability to be strongly overdoped should be valuable too, since underdoped cuprate grain boundaries are widely believed to be weakly linked. Accordingly, we here study property changes after oxygen underdoping the optimized, overdoped wire. While $J_c$ and vortex pinning diminish significantly in underdoped wires, we were not able to develop the prominent weak-link signature (a hysteretic $J_c$(H) characteristic) evident in even the very best Bi-2223 tapes with a $sim$ 15$^{circ}$ FWHM uniaxial texture. We attribute the high $J_c$ and lack of weak link signature in our Bi-2212 round wires to the high-aspect ratio, large-grain, basal-plane-faced grain morphology produced by partial-melt processing of Bi-2212 which enables $c$-axis Brick-Wall current flow when $ab$-plane transport is blocked. We conclude that the presently optimized biaxial texture of Bi-2212 intrinsically constitutes a strongly coupled current path, regardless of its oxygen doping state.
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