The band gap problem in DFT is not just due to approximate exchange-correlation functionals--it is a reflection of the fact that the Kohn-Sham (K-S) orbitals are a mathematical construction of a non-physical, non-interacting system of electrons that yields the true ground state charge density of the real many-body system. In exact DFT, the derivative of the total energy vs. number of electrons added to the system is piece-wise continuous, linear for fractional numbers added and with a discontinuity at each integer number of electrons. This discontinuity is a significant contribution to the difference between the true gap and the K-S gap. Hybrid functionals improve the treatment of band gaps because they incorporate part of the derivative discontinuity into the K-S eigenvalue gap (DFT+U also does this). You can read up on many papers on this subject. A lot of fundamental work was done in this area by Sham, Perdew, Burke, Cohen, Levy and others. I've included a short bibliography. This is definitely not the same as "rigid shift" or "scissor operator" which you mentioned.
Hybrid functionals can definitely change the character of the bands. At the end of the day you are still removing some of the spurious self-interaction of DFT exchange functionals. This should also give energies (e.g. formation energy, adsorption energy, etc.) that are more accurate, especially when dealing with systems that have more localized states that exacerbate the self-interaction issue. This is in fact the original intention of hybrid functionals: to correct the self-interaction present in DFT approximate exchange and get better predictions of the thermochemical properties of molecules. It's worth reading Becke's original hybrid functional paper.
That being said, I'm sometimes surprised by how little the general picture changes sometimes. One nice paper to illustrate this is this one by He and Franchini, which studies first-row transition metal perovskites with HSE. HSE can help split up manifolds that are normally tangled together in PBE, but the general picture of bonding stays pretty similar overall. This is in contrast to using DFT+U empirically and just fitting to a desired quantity, which can also change the bonding character significantly. Of course, in materials where self-interaction error is very large, or in some more exotic situations like orbital ordering, etc. HSE can still have a big impact on the electronic structure (see the He and Franchini paper sections on LaTiO3 or LaVO3 as opposed to LaScO3, or LaFeO3-- the electronic structures are significantly different with HSE vs. PBE in the former cases, and not as much in the latter with the exception of the gap).
- Perdew, J. P., Parr, R. G., Levy, M. & Balduz, J. L. Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy. Phys. Rev. Lett. 49, 1691–1694 (1982).
- Perdew, J. & Levy, M. Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities. Physical Review Letters 51, 1884–1887 (1983).
- Sham, L. & Schlüter, M. Density-functional theory of the band gap. Physical Review B 32, 3883–3889 (1985).
- Lannoo, M., Schlüter, M. & Sham, L. Calculation of the Kohn-Sham potential and its discontinuity for a model-semiconductor. Physical Review B 32, 3890–3899 (1985).
- Perdew, J. P. Density functional theory and the band gap problem. International Journal of Quantum Chemistry 28, 497–523 (1986).
- Becke, A. D. A new mixing of Hartree–Fock and local density‐functional theories. The Journal of Chemical Physics 98, 1372–1377 (1993).
- Seidl, A., Görling, A., Vogl, P., Majewski, J. A. & Levy, M. Generalized Kohn-Sham schemes and the band-gap problem. Physical Review B 53, 3764–3774 (1996).
- Perdew, J. P., Ernzerhof, M. & Burke, K. Rationale for mixing exact exchange with density functional approximations. The Journal of Chemical Physics 105, 9982 (1996).
- Cohen, A. J., Mori-Sánchez, P. & Yang, W. Fractional charge perspective on the band gap in density-functional theory. Phys. Rev. B 77, 115123 (2008).
- Yang, W., Cohen, A. J. & Mori-Sánchez, P. Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory. The Journal of Chemical Physics 136, 204111 (2012).
- Mori-Sánchez, P. & J. Cohen, A. The derivative discontinuity of the exchange–correlation functional. Physical Chemistry Chemical Physics 16, 14378–14387 (2014).
- Himmetoglu, B., Floris, A., de Gironcoli, S. & Cococcioni, M. Hubbard-corrected DFT energy functionals: The LDA+U description of correlated systems. International Journal of Quantum Chemistry 114, 14–49 (2014).
- Perdew, J. P. et al. Understanding band gaps of solids in generalized Kohn–Sham theory. Proceedings of the National Academy of Sciences 114, 2801–2806 (2017).