RAW answer seems to be "it works".

RAI is probably "your steed shapechange according to its CR, not your level".
Alternatively, I'd be tempted to count your mount as a gear, and merge it with you if you shapechange into something that cannot mount it. The intend of this "spells that affect only you might also affect your magical steed" is to make it compatible with invisibility, teleportation, and protection spells that would usually only target you. In other words, "you and your steed somewhat form a single entity whenever it is convenient for spells". Following this logic, RAI could be "you choose whether your magical steed remain on the ground, merges into the new form, or is mounted by it."

As for RAF, if you shapechange into something that can mount a dragon, shapechanging your mount into a dragon is certainly cool. But using this newly created dragon as an independent animal companion you send attacking alone rather than an actual steed seems like an exploit of the RAF.