… if I cast Booming Blade, am I locked out of Extra Attack because I took a “Cast a Spell” action instead of an “Attack” action?
Yes. Extra Attack specifies that you must take the Attack action. With Booming Blade, you’re still taking the Cast a Spell action, the singular melee weapon attack is simply how the spell is cast, not an entire Attack action.
This is the same for all spells that require you to make a weapon attack as part of casting it. The only spells that kind of break this rule are the various “Smite” spells, but they’re worded like “When you next hit a creature with a melee weapon attack…”, and generally use a bonus action rather than an action, so you can still use your action for the Attack action.