There’s plenty of applications beyond using them as solvents, e.g. fuel injection in jet engines, Diesel, or rockets; cooling in power plants or rockets; drilling; power cycles, etc.
Regardless of why you get to supercritical conditions, once you have them, that's plenty of reason to care about what supercriticality does in industry.
You want high pressures to achieve a high efficiency of a process (gas turbine, rocket, Diesel), then it's more of a side effect that the pressure is supercritical - I think that's what you allude to.
However, you may also want your fuel jet to mix more efficiently, so absence of phase equilibrium with surface tension may be advantageous.
You may also want a working fluid in a power cycle where expansion through a turbine does not end in subcritical spray that destroys your turbine blades.
You may also want to have a heat exchanger that does not have a 'boiling crisis', i.e. a subcritical vapor film that drastically reduces your heat flux when you exceed a certain limit temperature: think about it, the situation when your system gets unnaturally hot, and you really want to get rid of the heat, is when the heat transfer collapses. Now, there are subtleties about whether or where this can still happen at supercritical conditions, but let's just say for high enough pressures, a distinct phase transition no longer occurs (somewhere beyond 3 to 10 times the fluid critical pressure).