Lunar Pilot

The reissue and the Apollo watch are entirely different objects

The watch Scott wore was a manually wound mechanical chronograph. The Lunar Pilot is a modern quartz watch quoting its appearance. They share a design and nothing else, and no modern Lunar Pilot has any connection to the Apollo programme beyond that. This distinction is worth stating because reissue watches sold at second hand acquire, over time, descriptions their manufacturers never gave them.

A smoothly sweeping seconds hand on a quartz Bulova is correct

The 262 kHz oscillator drives the seconds hand in increments fine enough to read as a continuous sweep. An examiner who takes a sweeping hand as proof of a mechanical movement will misidentify the watch, and one who takes a sweep on a watch specified as quartz as evidence of substitution will report a correct watch as wrong. Bulova has used high-frequency quartz across several lines and the behaviour recurs — it is a brand signature and should be read as one.