Other than occasional minimal efforts like having to be aware of the time difference when calling someone in, or traveling to, another time zone, how do the time zones cause a problem for most people?

Timezones weren't created to make the time in each locale match solar noon precisely. They were created to solve the problem of myriad local times which became problematic with the spread of the railroads. In fact zones meant that 12:00 noon wouldn't exactly match solar noon in every town within the zone, but was close enough while still addressing the difficulties of each town setting its own time.

Pilots all use UTC for flight planning, navigation, etc. to reduce the chance of accidents.

While communications are faster and more connected, the vast majority of what most people do in our daily lives is still local.

I'd say that the prevalence of technology makes it easier to deal with multiple time zones. My wristwatch lets me instantly scroll to check the local time all around the world. An app on my phone let's me do that plus even helps me plan a meeting with people in different zones. Easy as pie.

In fact even if there were a single time, traveling longhaul you'd still have to adjust your thinking "hmmm, let's see if I want to be up around dawn, here in Tokyo that's at XX:XX on the clock, as opposed to YY:YY on the clock like it is back home.", etc. Even under a single timezone there would still have to be an agreed-upon international date line in order to know when the date changes.

There are even more variations out there than the global time zones. People seem to be able to cope. For example Ethiopia still maintains a different calendar and clock system alongside the Gregorian calendar and clock scheme used elsewhere. Many religious dates are calculated using lunar or Julian calendars even when their followers might live most of their lives using the Gregorian calendar.