I have done a research comparing JPEG and JPEG-2000 and couldn't reach the same or better perceptual level of quality for JPEG-2000 compared to JPEG in extreme compressing levels (for web).
Although, announced as ~20+% better perceptual quality with the same size than original JPEG, with available tools to reconvert existing JPEGs or even lossless PNGs, original JPEG was still superior. JPEG-2000 managed to get arguably better results only for huge images which are not as widely used in web (Full HD and bigger).
ImageMagick, GraphicsMagick, OpenJPEG all showed identical results (I assume, due to usage of Jasper to encode JPEG-2000) and all lacked options for encoding. Using Kakadu or online/basic converters didn't help either. In current status quo, tools like imagemin with plugins, can provide much better quality JPEGs on output than JPEG-2000 when maximally compressed for web. So JPEG-2000 being useful mostly for Safari, doesn't get any point to be another format to encode since regular JPEG provides better results.
What do I do wrong and are there any other tools/tricks that have more advanced options for JPEG-2000 encoding to finally beat JPEG?
It's not just you. JPEG 2000 isn't that much better. It's difficult to encode well, and we've got more mature JPEG encoders nowadays (MozJPEG, Guetzli).
JPEG 2000 is wavelet-based, which gives it better scores in the PSNR metric. However, JPEG 2000 excels in exactly the thing PSNR metric prefers too much: blurring. It makes JPEG 2000 look great in low quality range, where the older JPEG breaks down. But it's not helpful in practice for higher-quality images. Wavelets also struggle with text and sharp edges, just like the older JPEG.
JPEG 2000 has very nice progressive loading, but Safari has stopped supporting progressive rendering at some point.
The only practical use for JPEG 2000 is delivering photographic-like images with alpha channel, because JPEG can't do that, and JPEG 2000 still wins compared to PNG.