No, that's right. I'm not sure where the confusion is coming from.
A cost is paid upon activation of the card. You
HAVE to do
EVERYTHING in the cost
EXACTLY, otherwise you can't even activate the card.
The effect happens during resolution of the card. The general rule of thumb is that you do as much of the effect as possible, and it doesn't matter if an effect says "send the card to the Graveyard" but another effect is active that removes it from play instead. The card is removed from play instead. You're just resolving the effect to the best of your abilities.
For Abyss Soldier, the discard is a cost. You
must discard 1 WATER monster from your hand
to the Graveyard when you activate this effect. The returning of the card on the field to the hand is the effect. If there was a card effect in play that said any cards returned to the hand get sent to the Graveyard instead, that's fine. The card Abyss Soldier returns to the hand would simply go to the Graveyard instead.
The entire text of
Advanced Ritual Art is the effect. There is no cost. Everything that is written happens when you resolve the card. And you resolve it to the best of your abilities.
And there's no such thing as a "fizzling" effect. It's either there or it disappears.