It is not like it is not tried. Wavelet compression never took off. I don't know if it is because the format of is just better, or there was never enough investment into those formats.
Even in still-image compression, the difference is noticeable --- I have some high-resolution PDFs containing JPEG2000 scanned images, and they take significantly longer to render than the equivalent containing JPEG images.
Similar techniques as the ones used to optimise DCTs could be used for wavelets. There has just not been a demand. The standardisation effort tends to swamp out all alternatives once the decision to choose the algorithm has been made. There's a huge amount of momentum behind the standard which makes alternatives very hard to pitch. This is part of the reason why the same basic technique has been in prevalent use for 20 odd years and nothing else has come into play