Charlaquin
Goblin Queen (She/Her/Hers)
I definitely don’t think they were intended to be cumbersome action economy puzzles. But I do think they were intended to be more than just flavor. As you observe, they allow casters to be disarmed. They also limit the caster’s number of free hands. Casters aren’t meant to be able to freely cast while dual-wielding or while using a two-handed weapon. Clerics and Paladins are meant to be able to cast freely while wielding a sword and shield, while other casters are not (hence the “emblem” holy symbol option). Bladesingers can’t use their sword as a spellcasting focus not to complicate the action economy, but to prevent them from casting freely while dial wielding or while using a sword and shield. There are common magic items that can allow bladesingers to do this, but hey require attunement.If they have the Dual Wielder feat, I think that's within the RAW. The feat never explicitly states that you can't draw one and stow a second or vice-versa. Technically, RAI is unclear, but I don't think that it's really that important here.
Personally, if I were a DM then I would rule that a Bladesinger could use any one-handed bladed weapon as an arcane focus, especially when bladesong is active. It's kind of the point of the subclass. It's also important to remember that focuses and spell component pouches are a flavor rule, not an intentional mechanical restriction. They exist to allow a spellcaster to be partially disarmed and to evoke narrative imagery, not to be cumbersome action economy puzzles in combat. And if the designers did intend them to be cumbersome action economy puzzles, then they're bad designers and they should feel bad.
All that said, a big part of the point of the dual-wielder feat is to get around these sorts of restrictions when dual-wielding, so I would absolutely allow it.