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Olin has a great answer, I just wanted to add some of my personal experience. I only started laying out PCBs three years ago, and I use KiCad. KiCad got curved traces support in test version 5.99 over two years ago, and I immediately started using that build because I was fixated on the aesthetics of curved traces. Using the curved traces took away the push and shove routing option, and I persisted still. In hindsight, I spent too much time to get a layout done that could've been routed in a fraction of the time with push and shove.

I also spent a month evaluating TopoR version 7, another layout software that boasts organic traces and a powerful, feature-rich auto router. They've done a great job with that software, but by the time I've set up all the constraints, and ran the algorithm for a few hours, the board could've been already done by hand using standard 45 degree traces. This is, however, on a relatively simple, non-HDI, no impedance control - all the things that could benefit from the advanced constraints and guided routing. I've done a bunch of testing, and while I really liked the smooth, organic traces, it just wasn't practical for my use case.

After all that, I am in a place where I don't use curved traces at all, and I don't see them becoming mainstream.