Having issues rendering a transparent project on Motion

To start out, why are you exporting this as a ProRes 4444 video clip when a template would probably be less than a couple of MB in size and you're using it in Final Cut?

The Lens Flare generator has a solid color background (black). It's always going to export as a solid.

For the generator, you could try adding a Luma Keyer to the Lens Flare... but... I don't think you'll be satisfied with the result.

If the Luma Keyer doesn't work for you, try Converting the project to a Title and add the Title Background Source. Make sure the Title Background and the Lens Flare are in the same group. Change the Lens Flare Blend Mode to Linear Dodge (recommended), Screen or Add (you could rig these selections to change as needed). Also, if the position of the Lens Flare is off center, open the Generator Tab and change the Width and Height to ... just double the values of whatever your project size is (won't matter performance-wise). Other considerations: publish the Opacity of the Lens Flare if it's "too much".

There may be other options...

I think Fox's answer is right on point. Assuming PhilCF did not change the Project>Background setting, and left it at the default, the issue is probably 100% the fact that the Lens Flare generator does not have an alpha that cuts out the flare. If you add a Lens Flare generator to an empty project and bring up the alpha view in the caverns (shift-A) you will see that the alpha is white for the whole layer, it does not isolate the flare. FWIW, elements like this don't work well with alpha channels, as Fox already pointed out, the result of a light source with a matte, such as a lens flare, rarely looks good by matting it.

Also note that the Lens Flare generator's blend mode is set to "Add" by default. This is because in the real world, lens flares happen in the lens elements, when trying to create a realistic lens flare you want to put it on top of everything in the layer stack to get a more accurate result. The Add blend is simply adding and since most of the generator space is black, it adds zero for those pixels and ends up looks like it is matted. This is not unique to Motion's built in lens flare generator, mFlare does the same thing, you need to set the blend mode to get the right look.

PhilCF, I would also go with the workflow Fox recommended if your target app is Final Cut, it's actually much more efficient. If you are going to Final Cut, convert your project to a Title or Generator. Converting to a title, will give you the option to add a placeholder (for the underlying video), if you convert to a generator, add a Drop Zone. Place the Placeholder or Drop Zone at the bottom of your layer stack, and publish.

If you need to take this to another software, I would do two exports, one with the animated lens flare only, one without the lens flare. The lens flare export would not need an alpha, but the export would to matte the elements . Then in the other software, just use the Add blend mode to put it together.

Check your Project settings in the Inspector.

You need to have Background: Transparent. I am guessing that something else (Solid or Environment) may be selected there.