It's not rocket science how one becomes a champion. A trainer just has to beat the current champion after beating the Elite Four. As shown in Alola, if there is no champion, a trainer still has to beat the Elite Four (chosen by the organizer) and either defeat the organizer of the league or the competing trainer going after the title.
Neither of these applied here. Iris was just stated to have gone around beating Dragon-type trainers and...that was apparently enough to kick Alder off without even mentioning she fought him or the Elite Four...er, Elite One, in this case. It was way too easy to mention she fought him, but why not do that now? It may be implied, but still, why not mention it? There wasn't nothing to lose. That was just incredibly dumb. (It's waaay dumber because Iris places a great deal of respect on Alder in the games, so of course the anime destroys that bridge and makes the hole under said bridge a chasm.)