A Bonus Action attack is not an Attack Action.
The Player's Handbook, Ch 9 talks about the actions you can take during the Action portion of your turn. One of those actions is the Attack Action.
According to Two-Weapon Fighting
When you take the Attack action...you can use a bonus action to attack....
If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.
So, when you take the Attack Action, you can use a Bonus Action to attack and make "a melee attack". Regardless of how many attacks you get when you take the Attack Action, you only get the one, single melee attack as a Bonus Action.
But, just to be extra sure, let's take a look at Bonus Actions themselves. Under the Your Turn section of the Player's Handbook (Ch 9) there are rules for how Bonus Actions work.
You can take a bonus action only when a special ability, spell, or
other feature of the game states that you can do something as a bonus
action. You otherwise don't have a bonus action to take.
You can take only one bonus action on your turn, so you must choose
which bonus action to use when you have more than one available.
Whenever a Bonus Action allows you to do something that is typically an Action, it specifically says what and how you can do it.
- For instance, the Rogue class feature Cunning Action allows a Rogue to:
...take a bonus action on each of your turns in combat. This action
can be used only to take the Dash, Disengage, or Hide action.
- Another example is the Cleric's Healing Word spell. Spell casting typically requires the Cast a Spell Action, yet here we have a spell with a cast time of a Bonus Action. Does that mean you can take the Cast a Spell Action to cast what ever you want and cast Healing Word as well?? No. According to the Spellcasting rules (PHB,Ch 10) when you want to cast a spell with a cast time of 1 Bonus Action:
You must use a bonus action on your turn to cast the spell, provided
that you haven't already taken a bonus action this turn. You can't
cast another spell during the same turn, except for a cantrip with a
casting time of 1 action.
Neither of the examples above are attacks. Since there are a variety of Bonus Actions available in the game, a Bonus Action is not automatically related to an attack.
Finally, you are not getting an Action as a bonus (small 'b'), but are instead triggering a Bonus Action which is its own unique entity and is not part of the normal action economy. Special game mechanics give you a Bonus Action which you can only use for specific purposes, you don't use a Bonus Action to take the Attack Action.
So, while I can aaaaaalmost see your logic, the rules clearly say that a Bonus Action gives an attack, not an Attack Action.