Leap seconds are trying to ensure that, standing at the point we arbitrarily picked for UTC at noon, the sun will be at its peak.
That doesn't feel like something that NEEDS sub-second precision. I mean - by walking from one side of a time zone to the other, you can have an hour or more of imprecision. https://i.ibb.co/r000S5s/caxxyddlsgp11.jpg
> Is there a single system currently in existence which can handle a leap minute?
Every system can handle a leap-hour, they happen twice a year in many countries, courtesy of DST.
So let's just wait until there's 15 minutes of error - a couple of millenia from now. Then move the UTC prime meridian about 3.75 degrees / 400km, fixing the error while keeping UTC monotonically increasing 1 second per second.
And to update our clocks, use the one mechanism that we already rely on comfortably - timezones. Add 15 minutes to each timezone, the same way we handle DST now.