That's helpful!
I've done a lot of work in audio synthesis, which is notoriously difficult measure. The gold-standard is human ratings of audio quality, but it is tough to design good tests (easy to fatigue raters) and the iteration time waiting for results is quite long.
Instead, there's now some projects which use neural networks trained on human ratings to predict audio quality, such as ViSQoL: https://github.com/google/visqol
This opens up fast iteration - scores going up generally corresponds to higher quality - followed by human testing at major milestones (eg, releasing a paper/model). VISQOL has a harder time comparing 'unrelated' models, IMO - ends up being not so great for comparison of different techniques, but excellent for measuring incremental improvement or catching regressions.
But, in the end, yes - you can use NN's to measure the quality of other NN's, so long as you're careful about it and make use of human raters from time to time as well.
The problem of test data getting into the training data seems to be an especially pernicious issue with LLM's, which isn't really arising in the audio synthesis space.