Does computational chemistry get the molecular orbitals of dioxygen wrong?
A triplet species, as your MO scheme shows, has two unpaired alpha spin electrons, which is why you don't have to look at the beta orbitals at all but you have to look, if the $\alpha$-HOMO and $\alpha$-HOMO-1 are equivalent or not.

After optimizing the geometry of triplet oxygen using the method you states, B3LYP 6-311G**, we end up with a bond length of $\pu{1.2058 Å}$ and are given the following energies:

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            Population analysis using the SCF density.

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 Orbital symmetries:
 Alpha Orbitals:
       Occupied  (SGG) (SGU) (SGG) (SGU) (PIU) (PIU) (SGG) (PIG)
                 (PIG)
       Virtual   (SGU) (PIU) (PIU) (SGU) (SGG) (SGG) (PIG) (PIG)
                 (SGU) (PIU) (PIU) (DLTG) (DLTG) (DLTU) (DLTU)
                 (SGG) (PIG) (PIG) (SGU) (PIU) (PIU) (SGG) (PIG)
                 (PIG) (SGU) (SGG) (SGU)
 Beta  Orbitals:
       Occupied  (SGG) (SGU) (SGG) (SGU) (SGG) (PIU) (PIU)
       Virtual   (PIG) (PIG) (SGU) (PIU) (PIU) (SGU) (SGG) (SGG)
                 (PIG) (PIG) (SGU) (PIU) (PIU) (DLTG) (DLTG) (SGG)
                 (DLTU) (DLTU) (PIG) (PIG) (SGU) (PIU) (PIU) (SGG)
                 (PIG) (PIG) (SGU) (SGG) (SGU)
 The electronic state is 3-SGG.
 Alpha  occ. eigenvalues --  -19.28009 -19.27992  -1.31462  -0.83929  -0.56862
 Alpha  occ. eigenvalues --   -0.56862  -0.55311  -0.31192  -0.31192   &lt;------
 Alpha virt. eigenvalues --    0.19733   0.66242   0.66242   0.69937   0.71138
 Alpha virt. eigenvalues --    0.77370   0.78062   0.78062   1.23343   2.23120
 Alpha virt. eigenvalues --    2.23120   2.53472   2.53472   2.80188   2.80188
 Alpha virt. eigenvalues --    2.83167   3.40715   3.40715   3.75295   4.50288
 Alpha virt. eigenvalues --    4.50288   4.64528   4.80485   4.80485   5.97229
 Alpha virt. eigenvalues --   49.48209  49.67839
  Beta  occ. eigenvalues --  -19.24954 -19.24923  -1.25730  -0.75086  -0.51327
  Beta  occ. eigenvalues --   -0.46530  -0.46530
  Beta virt. eigenvalues --   -0.11708  -0.11708   0.24099   0.70896   0.70896
  Beta virt. eigenvalues --    0.71935   0.72389   0.79954   0.83108   0.83108
  Beta virt. eigenvalues --    1.27043   2.27711   2.27711   2.62147   2.62147
  Beta virt. eigenvalues --    2.87800   2.89802   2.89802   3.46348   3.46348
  Beta virt. eigenvalues --    3.78751   4.58054   4.58054   4.66543   4.88322
  Beta virt. eigenvalues --    4.88322   6.00448  49.51237  49.70844


You can clearly see, that the $\alpha$-HOMO and $\alpha$-HOMO-1 are degenerate with both an energy of $\pu{-0.31192 Eh}$. If you look at both MOs you can easily see why, because they are qualitatively the same combination of 2p AOs but with different orientation.



As symmetry was turned on, I guess, this needs to be the case, but the result stays the same, if it is turned off.

So, does computational chemistry get the molecular orbitals of dioxygen wrong? No, the MOs are right.



But actually you have a different question: Why are the energies of alpha and beta MOs in unrestricted open shell calculations not identical?

There are two methods to treat open shell systems. One is restricted open (RO*) and the other is unrestricted (U*). To not expand it too much, the main difference is, that as restricted closed shell methods, the restricted open shell methods force the alpha and beta orbitals to share the same MOs. Then, of course, they also would have the same energies.

But the energies that you show are from an unsrestricted open shell calculation. If you use this method, alpha and beta MOs are not forced to be in the same spatial MOs. And this freedom leads to different energies alpha and beta MOs.