I'm trying to make a Sunlight themed Sorcerous Origin and wanted to add spells to it as a feature. I was able to put all the spells into the "Additional Specific Spells" section.

That seemed to let me learn the spells but didn't keep them as not counting against known spells and didn't keep them always "prepared" either. So I tried adding them to "Additional Spell List" under the subclass' spell list feature.

But that didn't work either, so I started adding them to the "Spells" section under the subclass feature. Now, that DID work. HOWEVER, I found I couldn't add three of the spells I wanted to add for some reason: Dawn, Searing Smite and Sickening Radiance. And I can't for the life of me figure out why. It's not because they're aren't Sorcerer spells; most of the spells here aren't Sorcerer spells. If it's because I don't own the source material that seems odd to me (especially since I DO have those two sources [PhB, Xanathar's] shared with me), especially since I could put the spells into those other two areas but not into this one specifically. Thoughts?

You need to own the spells (or their sources) on DDB in order to use them in your homebrew. Having them shared with you doesn't count. Your options are to purchase those spells, or to have the person sharing the PHB make the homebrew subclass for you.