How to keep the conservatory roof drain clean

Oh dear. That's an unfortunate detail. :( I'm afraid DA01 is actually correct, the proper solution is tear out and properly rebuild. The proper approach to this would have been to either stop the barge end (the gable end overhang) several inches above the lower roof so the bottom could be properly enclosed and the lower roof properly flashed. This would not be a good detail architecturally though.

Another approach would be to fully extend the barge end so that it is actually fully framed into the lower roof, then flashing applied around the intersection. The barge end trim could extend a bit over the flashing, stopping short of the lower roof an inch or so. But the main structure is actually physically connected to the lower roof.

Those are the correct options, which involves a lot of remedial work and undoing much of what has been done in this area. If need be, you could do a stop gap treatment that will seal off the gap for now. This is far from ideal, and proper treatment should certainly be done when it comes time to re-roof.

This would involve fitting carefully shaped wood shims into the gaps to give the appearance the barge end had been framed into the lower roof. The outward facing shims could be toenailed in place, but the parts under the soffit would be just a pressure fit or sort of glued in with caulking. There will likely still be small gaps that need to be filled with caulk. Use a paintable caulk so when it's done and painted, it will not appear to be a patch job from any distance.

I don;t like this final option at all, but I understand if you are unwilling to do a proper fix at this point and thus offer this suggestion.

This is a common mistake, which will lead to premature roof wear. What the installer should have done, was to install an A to B transition elbow, and continued down the roof with a section of down spout. Then used an A elbow to have the water dump directly into the lower gutter.

It should end up similar to this

Allowing the high volume of water to flow over the roof, will damage the shingles and cause them to wear prematurely. Even if you turn the spout adjacent to the slope, it will still cause damage to the roofing. Spreading the flow over the roof in this way, will not alleviate the damage that is caused. In either case, you're allowing all the water shed from the upper roof, to flow over a small section of the lower roof. It's a terrible idea, and should be avoided.

I'm not sure exactly when builders decided to try and save the $10, by not installing the downspout along the roof. But it makes me so sick every time I see it, I just want to climb up there and fix it myself.