Why does Antonio leave?
At the end of Nunn's "Twelfth Night" all the troublemakers leave Olivia's household.
Andrew Aguecheek, as a disappointed suitor, naturally has to go. He couldn't very well stay.
In other productions I've seen, Toby Belch and Maria (now Belch) remain (Olivia is related to him, after all, and she's put up with him this long). Perhaps they're just going on a long honeymoon. But they are divisive characters -- Maria can be particularly nasty and spiteful -- and I can see why Nunn chose to take them out.
Textually, it seems like Malvolio might be staying on (the way Hawthorne plays him, you wonder how the household will run without him). But he, too, is, in a way, a disappointed suitor. And his pride probably drives him from the house (unlike the others, he seems to be sneaking out)
But I don't understand why Antonio is leaving. Nicholas Farrell has been sympathetic since "Chariots of Fire", and his Horatio is the only character worth saving in Branagh's wonderful "Hamlet." Antonio has shown nothing but kindness to Sebastian, and Orsino seems to be forgiving him. Why doesn't Sebastian have him hang around for the party? The chap who saved Viola is there.
Since Nunn did not do a commentary on his decisions on this play, I have to rely on your helpful suppositions.