What Limits the power of a typical LED
Just going to the question of dissipation, diamond is probably where things are headed in the nearer term with LEDs. Diamond's thermal conductivity (usually treated by the phonon model for lattice vibrations) is about \$2200\:\frac{\textrm{W}}{\textrm{m}\cdot \textrm{K}}\$ -- which is many times that of copper or any other metal at ordinary temperatures and pressures and a hundred times as good as sapphire.

Look up DIW (diamond islands wafers), CVD (continuous vapor deposition), and DLC (diamond like carbon) to get an eye into where things are currently at, I suspect.

Also, licensing for high brightness LEDs are extremely narrow/tight. A few companies own key IP (Nichia's YAG phosphor, for example.) And they are locking things up pretty tight, right now. (I worked at OSRAM for a while. Since I know how to develop and manufacture new phosphors in heated quartz chambers, and have extensive datasets on the last 50 years of phosphor studies and development, I've considered getting back into this business... but the barriers are high and I'd need the right partners here.)

But like the old saying goes, "It's only as strong as its weakest link," transporting heat away is only as good as the poorest conductor in the chain. So you need to hook up good conduction with good conduction or the whole thing falls on its face. And even with diamond, the weakest link will still be right at the GaN itself that generates the light. So novel designs may need to work on "diamond like" emitting lattice structures, eventually. I'll look forward to seeing that development.