You probably are one of those who intuitively see circularly rounded corners brutal, but cannot tell exactly what the brutality means. Most of us see circularly rounded corners perfect. The arc joins tangentially to the line segments and nothing more regular curve exists than the circular arc!
So we have used to think including me. But your opinion can be understood by reasoning. Physicists can prove that circular rounding in a railway or in a funfair big dipper track would cause violent very unpleasant thumps when the straight part changes to the rounded one or back. The feeling of the thump is caused by the centrifugal force turning suddenly ON or OFF. Tangential joint is not good enough. To avoid those thumps the curvature(* changes also should be smooth. I guess the brutality of the circular rounding is the abrupt change of the curvature.
In theory there could exist a tool which changes circular rounding arcs to curves which join to the line segments tangentially and in addition have gradually changing curvature which is zero at the joints but the average of the curvature is the same as what the original circular rounding had. That makes the new rounding curve fit one degree more perfectly than a circular arc would do. So, you expect more than most of us, it's a total misunderstanding to claim you want less perfect roundings.
I have never seen such tool, but a curve math aware programmer could create it. Illustrator has commercial 3rd party extensions for ex. by Astute Graphics, but I haven't explored if there's also the more advanced rounding tool that you want. Their VectorSribe > PathScribe seems to make it, but I have no way to test it.
I must add that simply changing the corner points to smooth nodes makes the curvature changes continuous, but it harmfully also removes all of the straight line segments. Inserting the "round corners" effect leaves a part of the straight line segment intact.
If you have a shape which has only straight line segments the fastest way to get the corners rounded in the non-brutal way is to redraw the shape with the pen:
If you create new anchors by dragging the pen and do not try to edit the Bezier handles afterwards the handles stay symmetric. That means tangency and continuous curvature change.
To redraw a polyline lock it in the layers panel to prevent unwanted edits. Draw with the pen on it by dragging along the original polyline. The anchor points and the dragged handles snap if you have Smart Guides and Snap to Point options checked.
In the left a polyline waits for rounding. In the right the original is locked and a new curve is drawn on the original. The anchors are now shown without their handles. The image is from Inkscape, but the same method works in Illustrator, too.
One thing needs practicing: How to drag short enough Bezier handles in the fly. If they are too long the rounding looks bulged. Curve editing with the white arrow is fixes it, but that kills the speed of the process. Fortunately Undo works during one draws a curve. Ctrl+Z takes back the just inserted badly drawn anchor point.
*) The curvature means how fast the curve tangent direction changes. It can be measured as degrees or radians per unit length along the curve. Straight lines have zero curvature. Circles have constant curvatures 360 degrees per 2(Pi)R. That's equivalently as radians per unit length = 1/R where R is the radius of the circle. Graphic programs like Illustrator allow cubic Bezier curve paths which can join to straight lines without curvature discontinuity. The answer above presented one way to draw them. Every circular arc rounding of a corner of 2 non-parallel straight lines has always curvature jump 1/R.