I’ve had a Phillips “atomic” alarm clock beside my bed for 23 years that just sets based on radio signal and auto restores if the power is lost, to which it has a backup 9v battery. There’s a time zone setting offset slider and a DST slider. I basically touch it twice a year, maybe another if I move time zones. I’ve only had to touch it for DST (never switched time zones). Takes me conservatively 10 seconds to find a…
I did went GPS+ntp route. I have a few raspberry pis on my lan that serve NTP time. (you need more than one) Over the years I've experimented with various flavors. Right now I have a mix: 1) a pi3 with a gps hat and a battery backup for the pi itself. It uses ntp 127.127.20.20 for gps time, and 127.127.20.28 for PPS. The SMA antenna goes to receiver at the window 2) a pi3 with USB gps on a cable going to the window 3…
No. Accurate time is one of the prerequisites for acquiring a position fix, and it's the one that's the easiest and fastest to obtain from the sky. Typically a receiver should be able to get time within seconds from cold start if there's enough sky visibility, and once it does it is as accurate as it gets regardless of position fix status (unless you care about sub-nanosecond precision, but you need to be able to somehow consume that anyway and you won't with a Pi).
That said, NMEA output will usually contain the time from the module's internal clock that gets synchronized with GPS time when available rather than GPS time directly. Some modules can be configured to report when they have synced their time though, and you could pay attention to the time jumping to infer it too.