For civil purposes, I feel the second still makes more sense as 1/86400 of a solar day. Unless you have an atomic clock, you cannot accurately keep to the SI unit over weeks or months anyway, let alone the time between leap seconds.
Practically everyone has access to atomic clocks these days by the virtue of having dozens of them whizzing over our heads broadcasting their time and having handy-dandy receivers in our pockets. And even running your own atomic clock these days is not that far fetched, see open timecards for example.
And then you have sports, some of which are (or have been determined) on the level of 1/1000 of a second. I’m not sure how that level of precision goes hand in hand with that definition of seconds.