Right, leap seconds it is then, but note that that and minutes/hours/days/years here is something that is only required for human interfaces, while the computer itself only cares about a monotonically increasing seconds counter on its internal clock, which is required to be stopped/started/re-adjusted once in a while anyway.
Oh, and conversion between scientific and civil time is then the job of local polities, which they do for timezones anyway.
The current answer is everyone uses UTC, and they keep synchronized by observing the one universally disseminated time, which is UTC.
You use and synchronize to the same time standard as the world-metrologists do (TAI), the whole issue is that that's not the one that is being disseminated as the article and I already pointed out.
That would require either doubling the radio frequencies (and other channels) used for time broadcast, or changing the transmission format to include both (so not really separate). Some systems actually broadcast the current leap second count along with UTC, so TAI can easily be derived. Of course, that presumes that UTC and TAI keep being related by a leap second count. Going to fractional seconds of delta would again require changing protocols and data formats.
In addition, we’d presumably still like a civil-time second to be the same as an SI second, and not go back to the “rubber seconds” of the 1960s. But how then would we handle the fact that the solar day isn’t exactly 86400 SI seconds [0]? Add/remove some fractional seconds at the end of each civil day? That might cause even more issues in software than leap seconds.
[0] https://www.timeanddate.com/time/earth-rotation.html