Leap seconds are generally only inserted (or removed, as required, at least in theory) in the last minute of the last hour on the last day of December or of June (although any month is possible), and of course are there to align our clocks with astronomical time.

Astronomers are very interested in real and accurate time, so that they know where they're looking for things in the sky at any instant in time. What reference do we have for stars, if we can't relate their location to a particular angle from a location at a particular time of day on any specific day? And seconds from an epoch doesn't cut it. You simply can't say that a celestial event (e.g. a particular star passing a particular spot in the sky) occurred at 4:21:11.385 one year, but at 4:21:16.278 another year: we can't correct to the ultimate accuracy, but we can adjust our clocks to within a few milliseconds of correctness. They would have to make the leap-second adjustment anyway. And those with more specialized needs have the capability to correct even more accurately.

If we wanted to define time to please us geeks, surely we'd do better than to have 86,400 seconds in a day, or to define a second as so many oscillations of a cesium atom.

And it's not like an accurate clock is that difficult, relative to other solved problems. If leap seconds are causing grief, there's no hope for correct handling of daylight savings time. As for keeping track of leap seconds, there is room for a network service in concert with NTP serving up a list of all leap-seconds to date. And if a software designer doesn't understand that some minutes have 60 seconds, I would guess they have other serious bugs in their software too.

tf.nist.gov/pubs/bulletin/leapsecond.htm