We consider polynomial approximation on the unit sphere $\mathbb{S}^2=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2=1\}$ by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation, and hyperinterpolation); powers of the negative Laplace--Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical $t$-designs, which are point sets on the sphere which when used as equal-weight quadrature rules integrate all spherical polynomials up to degree $t$ exactly. More precisely, we use well conditioned spherical $t$-designs obtained in a previous paper by maximizing the determinants of the Gram matrices subject to the spherical design constraint. For $t\ge 2L$ and an approximating polynomial of degree $L$ it turns out that there is no linear algebra problem to be solved and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation, and filtered hyperinterpolation. For $t \in [L, 2L)$ the linear system needs to be solved numerically. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operator and well conditioned spherical $t$-designs can provide good polynomial approximation on the sphere, with or without the presence of data errors.