It's something people have strong opinions on. Obviously some people want to get rid of leap seconds but I personally like UTC with leap seconds. The only change I'd make is to require leap seconds to be announced several years in advance (which would require allowing a slightly bigger divergence between UTC and UT1). Computer systems should gradually migrate to using TAI internally, but a clock on the wall or a bus timetable should using something like UTC: something that, you know, stays in sync with the actual sun. Saying "oh we'll shift all the time zones around one day" isn't an acceptable solution as far as I'm concerned. And I don't believe leap minutes would work any better than leap seconds: in fact I'm fairly certain they would be a whole lot worse.
Anyway, that's an explanation of my opinion. Many people will disagree with it.
I don't think it's true that "changing time zones is something we already know how to do easily". At least not with the word "easily" in there. Time zones cause a lot of problems, which is why we don't change them constantly. And it would be nice to have a future in which time zones are changed even more infrequently rather than one in which they have to be changed to things like UTC-37:00 because they are specified in terms of a UTC that has become unmoored from the diurnal cycle of day and night.
But probably the main argument against getting rid of leap seconds from UTC is that we already have TAI for anyone who wants something like UTC but without the leap seconds. In fact we already have two things that are like UTC but without the leap seconds: there's TAI and there's also GPS time, offset from TAI by 19 seconds. If that's what you want, use TAI. Or use GPS. What's the points of adding a third kind of time that is just like those two but offset by yet another small constant offset?
I tend to think that there's a real use for something like UTC that stays in sync with the Earth's rotation (and thus with the diurnal cycle of day and night) and that if some coterie of bureaucrats turns UTC into another version of TAI/GPS then probably someone else will at some point have to invent another thing to replace UTC and then we'll be roughly back where we started just with extra confusing complexity.
Imagine you want to specify the date and time of an international online meeting. Today you'd probably use UTC for that. But if most of the world were more than 24 hours offset from UTC then using UTC would mean specifying the "wrong" date for the meeting. You probably wouldn't want to do that so instead you'd use whatever new thing has been invented to replace UTC, something that has leap seconds or leap minutes or leap hours (or leap days?) or some other way of staying in sync with the calendar. Leap seconds seems to me like the right granularity: they're small enough to be ignored for most everyday purposes, and frequent enough that programmers can't just pretend they don't happen.
(Sorry for the waffle ... I didn't intend to write that much. I should definitely care less about this seeing as whatever happens I'll be dead long before the shit hits the fan.)
It will take something like 5,000 years for solar time to drift from TAI by more than an hour. Given that time zones change a few times a year in random parts of the world (for example, eastern Kazakhstan changed from UTC+6 to UTC+5 to unify with the rest of the country this year), adding another occasion to change them every few thousand years makes no meaningful difference.
Your example of getting so far off that we have to change _days_ would take several times longer than the distance between the invention of agriculture and now. Honestly, I would be extremely surprised if human civilization survives that long, but even if it does, our descendants can just deal with the problem then. There's no reason to make our lives more difficult _now_ just to deal with this very niche hypothetical scenario.
Anyway, that's an explanation of my opinion. Many people will disagree with it.