Twisted bilayer graphene (TBG) hosts nearly flat bands with narrow bandwidths of a few meV at certain {em magic} twist angles. Here we show that in twisted gapped Dirac material bilayers, or massive twisted bilayer graphenes (MTBG), isolated nearly flat bands below a threshold bandwidth $W_c$ are expected for continuous small twist angles up to a critical $theta_c$ depending on the flatness of the original bands and the interlayer coupling strength. Narrow bandwidths of $W lesssim $30 meV are expected for $theta lesssim 3^{circ} $ for twisted Dirac materials with intrinsic gaps of $sim 2$ eV that finds realization in monolayers of gapped transition metal dichalcogenides (TMDC), silicon carbide (SiC) among others, and even narrower bandwidths in hexagonal boron nitride (BN) whose gaps are $sim 5$ eV, while twisted graphene systems with smaller gaps of a few tens of meV, e.g. due to alignment with hexagonal boron nitride, show vestiges of the magic angles behavior in the bandwidth evolution. The phase diagram of finite valley Chern numbers of the isolated moire bands expands with increasing difference between the sublattice selective interlayer tunneling parameters. The valley contrasting circular dichroism for interband optical transitions is constructive near $0^{circ}$ and destructive near $60^{circ}$ alignments and can be tuned through electric field and gate driven polarization of the mini-valleys. Combining massive Dirac materials with various intrinsic gaps, Fermi velocities, interlayer tunneling strengths suggests optimistic prospects of increasing $theta_c$ and achieving correlated states with large $U/W$ effective interaction versus bandwidth ratios.
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