Does Deathknell trigger when the unit dies after playing Smite on it, and thus banishing it instead?

No, Deathknell will not trigger if a unit is banished by Smite.

According to RiftJudge FAQ #9664 and Rule 808.1.d.1, the Deathknell keyword specifically requires the permanent to be Killed and sent to the Trash. Because Smite contains a replacement effect that banishes the unit instead of letting it die, the unit is never sent to the trash. Since the condition "sent to the Trash" is not met, the Deathknell ability cannot trigger.