Injecting a corticosteroid to treat a trigger finger is the usual and most commonly accepted treatment. For many people, the one injection is all they need. However, not everyone responds the same way, as you can see with your husband. A second injection has a lower success rate than the first one, and there is no proof that a third injection is of any use.
Because the trigger finger returned so quickly after the second injection, his doctor likely feels that a third injection just won’t help. That is probably why he suggested surgery, which has a very high success rate.