To illustrate, here is an image with gain turned down and an input transform LUT arri to rec709 applied as a global setting:

- With an input LUT
- input00086400.jpg (294.17 KiB) Viewed 8075 times
And here is the same correction but with lut applied as an output LUT, also as a global setting:

- Output LUT
- output00086400.jpg (294.82 KiB) Viewed 8075 times
See how different the highlights are. In the case of the input lut there is no way to bring them back. Lowering the exposure just makes them a uniform gray area as opposed a uniform white area.
As a somewhat helpful analogy I would say an input LUT in this case makes footage look like it was shot with a standard rec.709 camera. And then you grade.
In case of the output LUT it is if it still being shot with a rec.709 HD cam for the final delivery, but when you grade you change the light before it gets to this virtual camera. Difference being that unlike changing the light of the real world scene, you do so as interpreted by alexa and its color space.
If you would like the same effect on a clip by clip basis, you may right click on any node and apply a LUT there.
Input LUT option there is that lets say you have to quickly process a bunch of files, or if you might have a custom look LUT that was used for onset monitoring and the creatives would like to have dailys on a non-LUTtable devices with the same look.
Output lut is there to serve many other purposes, like conforming to a specific projection device, or as a more flexible option for getting into a ballpark for rec.709 deliverables in the case of the particular LUT we are discussing.
One can get the image to look good without the use of LUT, it really just depends on your workflow.