Lot of people seem to miss the point of the article.
Suppose you had a clock that counted seconds (in the way we understand seconds, moving forward one unit per second). If you looked at it in a few days at midnight UTC on NYE (according to any clock), it would not be a multiple of 86400 (number of seconds per day). It would be off by some 29 seconds due to leap seconds. In that way, Unix time is not seconds since the epoch.
You have it backwards. If you look at it at midnight UTC (on any day, not just NYE) it WOULD be an exact multiple of 86400. (Try it and see.)
Because of leap seconds, this is wrong. Midnight UTC tonight is in fact NOT a multiple of 86,400 real, physical seconds since midnight UTC on 1970-01-01.
He didn't have it backwards, he was saying the same thing as you. He said, "suppose you had a clock that counted seconds," then described how it would work (it would be a non-multiple) if that was the case, which it isn't. You ignored that his description of the behavior was part of a hypothetical and not meant to describe how it actually behaves.
Suppose you had a clock that counted seconds (in the way we understand seconds, moving forward one unit per second). If you looked at it in a few days at midnight UTC on NYE (according to any clock), it would not be a multiple of 86400 (number of seconds per day). It would be off by some 29 seconds due to leap seconds. In that way, Unix time is not seconds since the epoch.