And all it takes to store a time like that is a 64-bit integer and it is very convenient. And a lot of software do precisely that. Most timestamps are just that: they don't care about "real world" details like leap-seconds, {23,24,25}hours per day, etc.
Because, in many case, you really don't give a flying shit about the "real-world" time.
What is your point? That a server that needs to open trade precisely at 9am has to take leap seconds into account? Sure. The article ain't disputing that.
But a lot of application basically only need timestamps and everything becomes much simpler when one starts to reason in milliseconds since the epoch instead of "real world" time.
Btw... Many of your "real issues" are due to programmers not realizing that they could have realized the exact same system but way more resilient had they knew to use timestamps instead of "real world" time.
There was an amazing paper by Google on the subject by the way, where they explained how they were reconciliating their hundreds of thousands of servers' time by allowing it to be "not correct" over x minutes / hours to dodge many issues.
And they clearly did emphasize that: a) these problems shouldn't have occured in the first place and were due to poorly designed programs (poorly designed in that they relied on real-world time instead of simply internally using milliseconds since the epoch) and b) most programs on earth do not give a flying ^^^^ if a time displayed to the user is one third of a second off...