In finite elements, we generally use Gauss elimination or Newton-Raphson for solving the system of equations. Often the poor quality of mesh leads to an ill-conditioned global stiffness matrix or sometimes a degenerate element stiffness matrix. While I understand that least-squares is used when we have an over-determined system. The least-squares can certainly be used in finite elements. https://amath.colorado.edu/pub/fosls/hyp1.pdf
Do you think least-squares in finite elements can fix problems that arises due to poor-mesh quality?
I'm not aware of such kind of application for LS in FEM, but of course the problem arising from a poor quality of the mesh, which is an unavoidable approximation for a certain physical problem, can't be overcome throughout a different method to solve numerically the mathematical problem which is affected by a different kind of approximation. Therefore, for that reason the answer is no, least-squares in finite elements can't fix problems that arises due to poor-mesh quality.