It occurs to me you could skip the whole dragon sex thing, and have half-dragons be something that (eventually) happens to any creature that stays in the area that surrounds an adult (or older) dragon's lair for a significant amount of time (and somehow doesn't get eaten). If that is too big a power up for dragons, it could also be a ritual the dragon does ("I'm feeling like a nap, so I need some guards while I sleep for the next decade. Not any guards will do...."). Joe Peasant could differentiate between dragonborn and half-dragons (if he was foolish enough to hang around instead of run away), because bipedal half-dragons might be hanging around bulls with big dragon teeth, scaly wolves, and don't forget weird dragon/giant spider things....
That might be a good gimmick for kobolds. If the tribe pleases a dragon, they might become half-dragon kobolds.
Well traditionally half-dragons would have been the offspring of a dragon in humanoid form. So no “dragon sex” and the dragon would always be the father (because the human/elf, whatever is giving birth), and only dragons that can take a humanoid form without a spell could do so, making most of the metallic dragon offspring.
In Krynn (Dragonlance), draconians (the first half-dragons/Dragonborn in D&D) were the result of a magical corruption of metallic dragon eggs. There were five types, depending on the type of dragon egg.
Of course, eventually they wanted dragon-people that weren’t evil offspring from a specific campaign, and a different type for each metallic and chromatic dragon.
Dragonborn would be the dragon equivalent of lizardfolk, or the various fish folk. They aren’t a half-spawn, they are their own race entirely.
This might make a difference in different generation. For example, a half-elf/half-elf child is a half elf, but if one parent is human, the child is human with some elven traits, and if one parent is elven, then the child is elven, but perhaps some sign of their human heritage. Although they may still be considered “not elven” by other elves.
But something like a Dragonborn might only be able to have children with other dragonborn.