The problem goes beyond testing: your training might (1) be deprived of information your original data had, e.g., among a 1000 features, the key to the classification of interest might be one feature; how does your generator know to not distort this potentially at the cost of distorting the other 999 features? (2) might latch onto the assumptions made by the data generator, e.g., for a continuous valued feature, your model might bias itself towards the distribution moments that the generator assumed.
I am not sure generating good fake data is a problem different from good density determination. And in some cases, you might need to specify what parts of the original distribution you don't want the generator to mess with: consider an NLP dataset where your model must rely on sentence structure. Generating the right bag-of-words features might not help here: sequence matters. Or if you wanted to use contextual embeddings; sequence matters then too.
Even if you did manage to generate a "distributionally-compatible" version of the data, for cases where you perform some kind of data enrichment at a later stage, you could run into problems. For ex: if the original data has zipcodes that you wanted to mask, and your data generator substitutes them with arbitrary strings, then at a later point you cannot introduce a feature that measures the proximity of two locations.
Theoretically speaking if the generated data has the same distribution and parameters as the real data [1], and encodes similar nonparametric features like seasonality and user activity, I think generated data might be fine. [2]
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1. Admittedly tricky if you have limited data and no insight into the underlying population distribution/features, just those of the sample. But then you have a worse problem for modeling diagnostics anyway.
2. In the sense that anything is "fine", which is a spectrum that requires some critical skepticism in statistics. There are always caveats but it may still be robust and useful.
So you could instead test on fake data that has the same (or as much as possible the same) statistical properties as the data that you would like to use.