But it's not what defines the class.
But it does help. As an example, if we took all the features of the Cleric, and gave them the Wizard Spell List, then they're not really a Cleric. Without access to things like Bless and Divine Favor and Healing, we now have a different class feel, so their spell choice is important.
The same as if we took a Cleric and their own Spell List ... but prepared nothing but combat spells (as they technically "know" every spell on their list). That alone, with the same spell list, is a different type of Cleric than one who takes all healing and support spells. These two characters, even without their Class Features, are predefining their themes and roles. They feel different. The fact that they have this choice is a defining element of the Cleric (Do you want to be a Healer, or a Crusader?)
Spell choice and selection is a huge way to define "I who does what", just as much as Class Features were. If we took the Sorcerer's Font of Magic and Metamagic and all that and gave them the Druid Spell List, that, to me, is not a Sorcerer, It's some type of ... like, Dryad-creature or a living spirit of Nature. Rather than one who communes with and becomes a Force of Nature through forming a deep connection, this strikes me as someone literally touched by the very soul of the land to be its chosen in some way, or someone who is like the reincarnation of an ancient forest spirit, something like that.
Very cool flavor for a Sorcerer ... but not a Sorcerer. Spell Selection and Spells Known do define the way you play a class, by their very nature they have to because it changes your options and abilities.
I feel it's much the same way that a Fighter is going to be defined by whether she grabs the Bow or the Greatsword, or the Rogue is going to be seen differently if they take Acrobatics, Intimidate, and Knowledge History as opposed to Stealth, Deception, and Sleight of Hand. Though the same class, these are different approaches. The Sorcerer should have the same leeway to grab different spells and define whether they are a Greatsword Sorcerer or a Bow Sorcerer (obviously not using the weapons, just using the metaphor).
As it stands, again compounding the limited spells known and limited spell list, a Sorcerer just does not feel like it has the freedom to make meaningful choices to set it apart from other Sorcerers.
Spells Known does define a Class: Druid who uses the Wizard Spell List is not a Druid, and a Sorcerer who does not have access to some of the more impressive Arcane displays that a Wizard has feels gimped in some way.