A datetime with a timezone and a datetime without one are two different things, both of them useful. My birthday does not have a time zone. My deadline does.
The company deadline for getting some document returned? It might or might not, that's policy.
Poetically: we are born free of time zones. We die bound to one.
That seems subjective
A birthday doesn't have a time zone because the concept of a birthday is more about the date on the calendar on the wall, not any universally understood singular instant in time; and so what matters most when it comes to your birthday is where you are. Your birthday doesn't move to the day before or after just because you travel to the other side of the globe.
A deadline has a time zone because when your boss says he wants the project done by 4PM, he means 4PM wherever you both currently are -- and the specific instant in time he referred to doesn't change if you get on a train and move a time zone over before that 4PM occurs.
And it may in fact be time zone and not just UTC with an offset; because if your boss tells you he wants a certain report on his desk by 4PM every day; when your local time zone goes into daylight saving time, it doesn't suddenly mean the report needs to be filed by 5PM instead.
In the first of these cases, the date itself has no time zone and is valid in whatever context its being read from. In the second, the instant in time might be expressed in UTC time with or without a specific offset. In the third, each of the successive instants in time may shift around with respect to UTC even while it continues to be referred to with one constant expression.
None of these are subjective interpretations. They're a consequence of the fact that as humans we've overloaded our representation of date/time with multiple meanings.
Therefore, birthdays are not bound by timezone at all.
I don't get a happy birthday from customs though, cool!
For example: your local mom and pop corner store's daily opening and closing times. Storing those in UTC is not correct because mom and pop don't open and close their store based on UTC time. They open and close it based on the local time zone.
If you stored it as a local time (ie: with TZ), then if it's ever later translated to a different local time (different TZ), you're now dealing with all the quirks of 2 different timezones. It's great way to be off by some multiple of 15 minutes, or even a day or two!
Heck, even if it's the same exact location, storing in local time can still require conversion if that location uses daylight savings! You're never safe from needing to adapt to timezones, so storing datetimes in the most universal format is pretty much always the best thing to do.