Self-Referencing Effects when a Monster is Destroyed
Banisher of the Radiance wouldn't banish itself, it would go to the Graveyard.  Banisher of the Radiance has a Continuous Effects, which is only active while it remains on the field face-up. (Rule book page 14).

Continuous Effect
This effect is active while the Effect Monster Card is face-up on the field. The effect starts when the face-up monster appears on the field, and ends once that monster is gone or is no longer face-up; there is no trigger for its activation. These monsters are most useful if you have a strategy to protect them while they are on the field.

This is supported by previous rulings:

If &quot;Banisher of the Radiance&quot; and &quot;Sangan&quot; are destroyed simultaneously by an effect like &quot;Lightning Vortex&quot;, then both are sent to the Graveyard and &quot;Sangan's&quot; effect activates normally.[5]

Destroy means to send a card from the field to the Graveyard. Banisher of the Radiance prevents that from happening, so the hypothetical Card that you described would have no effect.

Destroy - A card is destroyed when it is sent to the Graveyard due to battle between monsters or by an effect that destroys a card. [...]

(Note: These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami.[6] They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise.)