You have a misconception about how things work in Dominion. That misconception is kinda 'common'. Players think about trashing the Pillage (or Embargo, or Feast) as the 'cost' of their respective effects, so when that trashing happens any other way, they think they are entitled to the effect. But that's not how it works.
From a game-balancing point the trashing might have been considered a balancing measure for the effect. Consider at one hand Feast
, which costs $4 and gains you a card costing up to $5, but trashes itself in the process. OTOH there is Artisan
, which also gains you a card costing up $5, but doesn't trash itself. It costs $6. It's clear that the trashing of Feast is meant to balance it only costing $4. From a design perspective, the trashing of Feast may be a cost, but from a rules perspective, it is just an effect.
Back to the actual question: the default for any text on an Action card is that it applies when you play the card. If it applies to anything else, it will specifically mention it, under a dividing line.
Take for example a look at Cultist
That card lets you draw 3 cards when you trash it, for example with Farmland. But note that it specifially mentions this condition ("When you trash this") and it is put under a dividing line, to make sure players understand that it is a separate ability, and not related to the "on-play" effect (draw 2 cards + more).
The card does not say you need to play it first.
is backwards. You need to provide a reason why you think you should follow instructions on a card. Otherwise, why not follow the instructions on Smithy anytime it is in your hand (without playing it)?
The reason you are giving is incorrect, because it is not based on any rule given anywhere. There is no rule that says "If a card's text starts with a certain effect, and that effect happens in the game by whatever means, you are allowed to continue to execute the remainder of that card's text."