It's a line to not cross. Sure, it starts with a ring into a volcano, but it doesn't end there. Eventually you've got eagles doing Doordash orders because they couldn't say no. Not a great look for a proud species.

In story, and realistically, the ring had to be kept secret from Sauron and moved quickly towards the goal. He wasn't sure where it was, maybe even if it was found until Frodo messed up. Eagles heading to Mordor? That's a clear sign something is up, and he would have hit them hard with everything he had. Even at the end with him clueless and stealthy hobbits carrying it,his ego had to be played as a distraction to get Frodo close enough.