Some of the points are valid concerns (no system will be perfect as long as we rely on the sun as being significant, which I'm expecting to be pretty normal for quite a while), but the author overcomplicates and overemphasizes the initial point.
"what time is it there?" isn't trying to figure out what time it is, it's trying to say "Should I expect my uncle to be awake?" _and_ then has to rely on a combination of google and internal expectations about a "normal" day to make that determination. In a single zone, you instead need to know how long the sun has been up there, which "4am" is just a shorthand to answer. Obviously if "4am" starts answering "how far away from now is it" (which the current system doesn't answer) and stops answering ("how long since sunrise") then we'll adopt another system to answer that second question. Given that the author literally consults a computer to manage our current system, doing so to get the answer in a new system doesn't sound any harder (and we'd quickly adopt a shorthand that we don't use today because we have nowhere to use it).
All other complaints build from that one. I don't want to minimize the complications of dealing with "days", but at the same time it's crazy to pretend that we're doing it just fine now. It fails ALL THE TIME now. Package shipping, news articles, etc. Heck, Fake News stories for April 1 now span a 36-48 hour period.
Pretending the current system works better just because we're used to the quirks is a false premise. Saying it would hard to switch is valid, but it's a short time high cost vs a perpetual but lower cost. Look at the US and the metric system, or recognizing that you can't find a reasonable definition of "planet" and include Pluto. (Okay, that last doesn't really have costs the way we lose multi-million dollar probes or smaller but more common errors in measurement conversions, but it does show how much resistance people will put up to avoid temporary discomfort).
Does anyone think we'll get LESS global? Does anyone think our current "day"-based system will work when there is more than a colony on Mars? (That may be 50 -100 years out, but I'm point the direction of the trend). At some point sunlight-based systems cost way more than we get for it. If someone doesn't think we're there yet, that doesn't mean the problem won't happen, so these same issues will still exist when we get there (in their mind).