Since HTML emails were also the rage, and email clients updated much slower than browsers, this technique was used well into the 2010s, if you wanted your thing to be displayed correctly.
I think Apple popularized the return of rounded corners in software with the iPhone X launch, when rounding the corners of smartphone screens became popular.
macOS rounded the window corners more, I think with Big Sur. And Microsoft followed suit in Windows 11.
Before using tables for rounded corners, I remember playing with a multi-chat app called Trillian (which handled yahoo chat as well as ICQ and others). It could be skinned to your liking in much the same way you were dealing with rounded corners. The top and side cells were repeating images so resizing the window would work well.
The great an mighty Oz's crowning achievement. Ok, maybe the iPhone was bigger.
And drop shadows!
I remember slicing and dicing up the rounded corners in photoshop.
No no, the middle cell contained the tiled background image, of course!