How about instead of setting the time to 23:59:60, the value 23:59:59 happens twice. When we have DST, and the time falls back an hour, we don't switch to some odd non-existant number for an hour so that we don't have overlap. We just set the clocks back to 1 AM. So all the times between 1 AM and 2 AM happen twice when switching off daylight savings.
The times between 1 AM and 2 AM don't really happen twice on the day daylight time ends, they are simply ambiguous unless daylight or standard time is specified. (In other words, you don't know which of 2 possible seconds 1:42:42 AM refers to until daylight or standard time is specified.) Similarly, your proposal would make 23:59:59 ambiguous without some additional specifier, in which case why not just use 23:59:60?
There is no one perfect solution, which is why there are multiple time standards, including TAI and GPS which do not incorporate leap seconds.
Many programmers are ignorant as to the more subtle aspects of timekeeping.
It does not help that for many programs, it simply doesn't matter.
It is also very easy to slip assumptions that are broken by leap seconds into code. (Every minute has 60 seconds--wrong! Every hour has 3600 seconds--bzzt! Every day has 86400 seconds--fail!)
