It's not only about display. If you store a user's appointment only as a UTC timestamp, you actually can't know the hour of the day (and the day itself to be precise) on which this appointment should happen, for a given calendar (probably the user's calendar, in a specific non-UTC timezone). You would have to guess by using the calendar's/user's timezone and compute some offset with UTC. But what if the user changes timezones or the timezone itself changes its value? Store without a timezone, and you know the exact hour and day the user intended. One is pointing to a day and hour in a calendar, the other is pointing at a point on the line of a linear timeline.
You're still just talking about client conversion on the write instead of the read. A Unix epoch is a UTC timestamp. That number is not time-zone-less it's just in the default computer time zone.