Personally I'd like to see timezones disappear too. Why does 9am have to be morning in every country? What would be so difficult about everyone in the world sticking to one timezone? Hell we could go all in and start using star dates.
If they're going to fiddle around with things like that it would be best just to decouple entirely from the Mean Sun's schedule and adopt a universal time as you suggest.
Bizarrely all official communications and timetables in the Soviet Union were published in Moscow time, across eleven time zones. That made catching a local train in Siberia quite an arithmetic challenge.
Interestingly, though, China works on a single time zone. So for a small-scale case study in how this would work, go live in different parts of China for a year.
Making 2) easier does not change anything with regard to 1).
Time-zones are related to the longitude of the location You also need to account for the latitude, and the time of the year, because the pattern of day and night depends on all three factors.
And replace the wristwatch of everyone who doesn't have a digital watch? And wall clocks?
(My point being that many watches and clocks already don't give us the literal time as it's spoken, just a representation.)
Edit: although just to be clear, I don't actually think we should switch to a single time zone. It would cause all kinds of headaches. I just think if in some universe it were to happen, it would necessarily go along with a switch to 24h time.