I've seen too many ways storing everything as UTC goes wrong/gets confusing. Round-tripping at least UTC offset (which unfortunately Postgres `timestamptz` in this case does not do and is effectively the same as storing as UTC) gives you more debugging tools for events in the past and more opportunities to do the right thing for future events in worst cases (strange, unexpected DST shifts). The best option is if you can also round trip exact time zones such as IANA strings like "America/Los_Angeles". Even fewer databases support that natively right now (as a single optimized column).
When UTC offsets roundtrip you can do all your date math as if everything was in UTC, but still not lose information from the user about what time they thought an event occurred at or might next occur at.
t depends on the system. Keeping a time zone alongside a UTC timestamp may be useful in some cases.
For example, if your system is supposed to remind a user to do something, such as take a pill, and the user changes time zones while travelling, you may want the reminder to occur at 9:00 local time wherever they currently are. In that case, storing the original time zone together with the event may not be necessary, because the relevant time zone is the user's current one.
Everything depends on the context. For me, however, there is no doubt about one thing: I store timestamps in UTC. Whether I also store a time zone, and where I store it, depends on the system and its business requirements.