Brush with far greater softness beyond opaque centre how ?

Reading the OP again, I am wondering how @Merlin3 intends using this proposed soft brush?  If you use a soft in a layer  mask and feather the mask, you get a softer effect.  Hmmm...  You would not have a fully opaque core doing this.

If you define a brush from a black to white radial gradient, it does appear to be progressive .

This is the gradient

And this is the brush preset made from that gradient. Not an exact match — I used guides to mark the positions — but PD close, and it is still progressive with close to fully opaque centre and transparent edge.

If @Merlin3 tells us more about how he'd like to use his proposed preset, I wonder if we can come up with some sort of workaround.

You're right – the standard Photoshop approach for e.g. brushes existed long before the Camera Raw converter was developed (actually Lightroom's editing approach), and the interface was indeed designed by a different team, led by Andrei Herasimchuk. They were allowed to come up wiith new ideas for this application, even though it would break with Photoshop's (and even Adobe's)  conventions.

Now, how to get more control over the brush ?

Maybe it helps to simply decrease the Brush tool's overall Opacity by using its slider or numerical keys from 1 to 0 (keys in compliance with setting a layer's opacity, when using the Move tool).

But if you really need an opaque center with a different 'fall-off' grade, you could consider creating your own brush shapes. Create a circular black to white blob, or any other cloud, spatter, or textural drawing. This image will control the brush density ranging from 100 to 0%. Its size depends on the largest size and smoothness you need. Maybe 200 x 200 pixels is already enough. Select it with the marquee and use Edit – Define Brush to create a custom brush out of it. You can see it appearung in the Brushes window.

Take a look at the very first settings category on the left side in the Brush Settings window, to decrease the Spacing, if the brush seems to be staggering a bit.