I figure that has a lot to do with audience. The films were meant to appeal to an entire range of people, which meant being palatable to adults, who of course would find child sidekicks more than a little ridiculous. That's adults...but what about children and kids' entertainment? Robin was present in B:TAS and 'The Batman,' and, flying solo or not, he was still assumed to be Batman's former sidekick in Teen Titans.
If your intention is to have a serious film that the average adult won't be embarrassed to go see then sure, omitting Robin is a safe bet. The same applies if you want your comics to have an ultra serious, 'realistic' edge. If your intention is to write a story that is nostalgic or is not trying to separate itself from the comic books of the past, well, Robin works. The same goes for stories aimed at younger readers. I think there is obviously still a place for Robin, as long as we're willing to embrace the (arguably inherent) 'childish' and whimsical aspect of superheroes. It has more to do with the writer's intention and reader's expectation than anything being outdated and wrong with the concept, imho.