As a whole, of course you have a point - big visualisations when done properly should help with data exploration. However, from my experience they rarely (but not never) do. I think it's specific to the type of data you work with and the visualisation you employ. Let me give an example.

Imagine we have some big data - like an OMIC dataset about chromatin modification differences between smokers and non-smokers. Genomes are large so one way to visualise might be to do a manhattan plot (mentioned here in another comment). Let's (hypothetically) say the pattern in the data is that chromatin in the vicinity of genes related to membrane functioning have more open chromatin marks in smokers compared to non smokers. A manhattan plot will not tell us that. And in order to be able to detect that in our visualisation we had to already know what we were looking for in the first place.

My point in this example is the following: in order to detect that we would have to know what to visualise first (i.e. visualise the genes related to membrane function separately from the rest). But then when we are looking for these kinds of associations - the visualisation becomes unnecessary. We can capture the comparison of interest with a single number (i.e. average difference between smokers vs non-smokers within this group of genes). And then we can test all kinds of associations by running a script with a for-loop in order to check all possible groups of genes we care about and return a number for each. It's much faster than visualisation. And then after this type of EDA is done, the picture would be produced as a result, displaying the effect and highlighting the insights.

I understand your point about visualisation being an indistinguishable part of EDA. But the example I provided above is much closer to my lived experience.