Yeah they can.
Wizards get two spells of their choice per level and have done since 3E. There's no requirement those spells come from like, the Wizard's correspondence course or something.
Bards and Sorcerers pull magic from entirely different sources to Wizards, but cast exactly the same spells. That actually hard-proves you're wrong. If arcane spells weren't "floating around the ether", a Sorcerer wouldn't cast Magic Missile, he'd fire an energy blast like a Pathfinder Kineticist or something. You can't even argue with that.