I don't think that's how the rules work nobledrake. Look at bard focus cantrips. You gain those spells in your repertoire without also gaining access. Class feats are actually the opposite and always tell you what the boundaries are for options (see every spell casting class feature limiting you to common spells or those you have access to).
Oddly, neither the Cavalier or Animal order druid feats which grant you an animal companion list a rarity limitation. Other bits in the rules are explicit that class features/feats or ancestry feats can give you uncommon options, but not rare ones (CRB pg 13, pg 488).
I don't understand what point you are trying to make with the bard focus spells.
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The reason all the class stuff is uncommon is explicitly so that you can't choose them without getting the specific Feat/option. There are options that are rare, but they're just that; Rare. Rain of Embers Stance is an option that's Rare, but is only available through a Monk Feat. Options have these tags so that the GM can know that they probably shouldn't just be given out to anyone who asks. Since there are no class feats/abilities that explicitly grant access to the Riding Drake, the answer is "Ask your GM". Maybe pitch them an idea of a story arc where you befriend this drake and it becomes your ally, to explain why you have it and why you're a cavalier.
If it's for PFS, and as it doesn't have an Access line, the only way you could have a Riding Drake is by spending Achievement Points in PFS to gain access to it as an option