Rounded wood corners were traditionally used for horsehair plaster on wood lath in the 19th century. There was a slight beveled groove at the vertical joint for wallpaper to cover the transition smoothly.

The modern version is plastic and can be tightly integrated into the drywall work and painted. It has a softer old fashioned look but still presents an awkward transition from round to square at the baseboard.

I was told that round corners were a thing of the 90's and early 00's. Apparently back to square is all the rage now. Everything is cyclical. Go with what you like and what works for you...chances are it'll be popular again in a couple years.

If they offered it in a smaller radius, it could look more historical; the 3/4"r. that is sold is too large by half. 1/2" r would look quite nice I believe. The commercially available plastic stuff is such a prominent look at me effect it's like postmodernism.
It's harder to install than metal, if all you do is metal.
Casey

Around here (mid-Canada) its an out of date thing.
But if you like it, do it.

Personally, I feel it's more of an outdated look. But if you prefer that look, then go with what you and DW like!

We did a tight radius "baby" bullnose at our outside corners. It cost a little more but doesn't read as a dated radius while presenting the practical properties of a radius.

Rather than just doing a square corner on the baseboard our baseboard corners all have 45 degree wraps which eases any awkwardness that Renovator mentions.