It’s uniform over the space of positive definite correlation matrices. In your plot of eta=1 you see that this is clearly not uniform across all values of correlation. The reason for this is that the space of PD correlation matrices concentrates around 0 as the dimension of the correlation matrix increases. Geometrically this is like all the volume is contained on the surface of the hypersphere.
You have some a prior information and so should incorporate it. The LKJ prior is great when you have no information about any of the correlations or the information you do have is on all of the correlations and not any one specifically. However, if you don’t want the bell shaped distribution when constraining the median correlation to be 0.1 or whatever then you have to do some different things. You can look at adding a bound like 0.3 and messing around with eta, putting bounds on is described at Constraints on LKJ prior - #12 by spinkney. That also allows you to constrain different correlations to be equal, which also includes making them into blocks that all are the same value.
If you suspect some correlations spike towards the pole (-1, 1) but many are small or 0 but you don’t know which. Well, there’s not too much to do today. I’m working on priors for this but my day job gets in the way of progress.