If research isn't profitable, there's something seriously wrong with the way research is being done.

Not necessarily. In many fields there's simply a very large gap between when an idea is discovered and when it is applied. Since most scientific research is either freely or cheaply available, and the applications happen many steps later in the process, scientific research functions as a public good, and thus has trouble deriving individual profit even as society benefits from it.

Lots of industries do fine without patents.

Many of those are industries with little innovation. Others rely on industry secrets. And others, like microchip manufacturing, rely on a constant arms race of new approaches so that even if one had the details of what someone else was doing, they will be already working on the next thing.