For a quote from Christopher Nolan as to why Robin is not in his trilogy (except the reference in the end of the third film):
“Dick Grayson’s still in a crib somewhere. I seriously doubt I will even be involved when Robin’s in the franchise.” – Christopher Nolan
Specifically (and I don't have a quote onhand at the moment), Nolan's films were patterned after Batman: Year One, where he's just starting out, and hasn't met Robin.
In comparison, the Batman of the Snyder films seems to be set in a post-Robin phase, after the death of Dick Grayson, which he considered pivotal in Batman's personality.
Rather than showing an ongoing partnership, Snyder's version of Robin was meant to be more about the idea of Batman's younger ally. Snyder's Batman, at his core, is a spiritually and psychologically broken man, with his arc being one of his journey back to the hero he was by way of Superman and the Justice League. More to the point, how much Bruce Wayne had invested himself in Grayson's upbringing and how deeply his loss affected him was at the heart of the story of Bruce Wayne being portrayed by Ben Affleck, with Robin playing an emotional role in his story rather than a literal one.
Had he been able to film all of the movies he originally planned, Snyder did plan to introduce Robin.
I'd always wanted to explore the death of Robin. And if there ever was going to be a next movie, which, of course, there probably won’t be, I wanted to do a thing where in flashbacks we learn how Robin died, how Joker killed him and burned down Wayne Manor, and that whole thing that happened between he and Bruce.
So in short, the recent films are set before, and after, the period of Robin's existence, explaining his absence.