In the following example, I have one wife, one daughter and one son.
(1) My wife, Angie; my son, Mike; and my daughter, Danielle, will be at the picnic.
I think the semicolons are very technically correct, but they are an eyesore.
Other possibilities:
(2) My wife, Angie, my son, Mike and my daughter, Danielle, will be at the picnic.
(3) My wife Angie, my son Mike, and my daughter Danielle will be at the picnic.
Which is preferred -- 1, 2 or 3 above?
All of them are technically defensible, though if I was to go with the first I'd favour either a semicolon after Danielle for the sake of internal consistency, or else no comma after that word.
The third is perhaps the least defensible because, as you say, it's considered proper by many to always use commas with non-restrictive apposition.
But this is not an absolute rule, and as it is it works like a "dialed-down" form of the first version.
And while it's the hardest to make a case for by reference to any stated rules, I think it's by far the cleanest.
(I might put a comma after Danielle here, but I might not—sometimes when thinking about these things its hard to know which of two reasonable options one would actually choose oneself if writing it itself—but that's another matter).
