So here is my shot.
I do have a sample build attached if anyone really want's to look at it, but it is just a case study and not the focus of the post. In the build I incorporate some of my own ideas to make a Drake that addresses a couple of the many concerns about them.
From a mechanical viability standpoint, I think that many of us can agree that as written, Drakes are by FAR not an optimal character choice. Their powers are cool but not game-breaking without a massive time and karma investment that would make any other character just as powerful. One way to help with this is to allow the use of the Draconic Magic Tradition from Forbidden Arcana. The unique benefit of this is that the Mage can summon any kind of spirit for any kind of role, and that the Drain Code is Willpower + Magic. The sample build shows how this can help a mage get even more mileage out of a Power Focus which can be pretty powergamey, but is just an example, not necessary by any means, and it only helps Mages so it is not a perfect solution.
Next is about making them more manageable and addressing the concern of "How do I be a Drake and not cause HTR teams to swarm my friends at every conceivable moment for a chance at my corpse?" This takes some work. Having a well-crafted backstory and some of the right positive qualities can really help I think. I will again use the attached character as an example. The skeleton of his backstory so far is that he was held captive and experimented on by someone, whether some Corporate research team that officially doesn't exist, a rogue magical group, or was being used as a renewable source of powerful reagents for some powerful but messed up Mage. Someone powerful found out and took pity on him and helped him escape. I'm thinking that this someone would be one of the Dragons that does not enslave Drakes or prefers to use them like they do Shadowrunners. I've heard that there are some Dragons like that but I am not too sure on the lore details myself. The Dragon that saved him then dropped him off in Seattle with the understanding that there will be times in which they have tasks (Runs) for the character, so they better prepare and get some experience.
This backstory, brief and basic as it is, addresses one example of how or why a Drake PC might come to be a Shadowrunner. Also, since he is sort of claimed by this Dragon, other Dragons and Corps are less likely to pull him off the street, while he still may be hunted by other groups or individuals. It also gives the group a potential hook into some high level jobs in the future.
Regarding Positive Qualities that may help, I have a few ideas as well. In order to have the Draconic Magic Tradition, you have to have a Dragon mentor, which is another reason why it would make sense for it to be a Dragon that saves him. The Sensei quality might be a good fit for a character like this because his "rescuer" can act as his sensei. They would certainly have some motivation to do so if they wanted the character to be useful to them. The Erased quality could also be very helpful to keep the character under the radar, and it would be trivial for a Dragon to have that done. Friends in High Places could help a lot. The character could take the Dragon as a contact as well since they are going to eventually have jobs for him and helped the character with the basics that they would need to get started in the Shadows as a Drake. The contact would work similarly to a Fixer, but not as available as the usual ones.
I took the Negative Qualities: Flashbacks at the 7 karma level, Insomnia at the 10 karma level, and Social Stress to represent the PTSD that the character has after his inhumane treatment.