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In what respect building a theory that explains an experiment that was not consistent with the then prevalent theories (https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper...) can be considered as solving a pseudo problem?

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The equations that students must often learn for special relativity were already known, hence the Lorentz naming. The question at the time was largely about how this undetectable substance called the ether behaved.

Another example might be the "discovery" of anti particles or of black holes. Both of which were long regarded as just mathematical curiosities.

Anti-particles had a very short life as a theoretical construct, Dirac predicted them in 1931 and Carl Anderson discovered the positron in 1932.

Also I'll add that there were many "outs" for those that supported aether theories such as frame dragging or simply experimental error as those themselves doing the experiments often believed and kept trying to develop more refined experiments to prove the existence of the aether even into the 20s

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