For may years we were happy to give positions to one minute of arc i.e. 1 nm. Very occasionally there was a requirement to refine it further e.g. bomb aiming, surveying etc. Surveyors and those with a need to use Norie's Tables have always used seconds of arc giving a resolution of 100yds and beyond. Aviators have generally used decimal parts of a minute, giving an initial accuracy of 600 yds to one place of decimal, or 60 yds to two places. Early GPS ramp charts included a Lat Long overlay with a decimal grid, it was easy to use but was originally only intended to give a refined position for the alignment of Inertial Navigation Platforms whose long term accuracy depended upon it.
dd mm.m is sufficiently accurate for most aviation functions, the introduction of seconds is both unnecessary, and as you have pointed out the source of confusion, and good material to keep Murphy's Law in business. I am always fascinated that pilots are obsessed with levels of waypoint accuracy that exceed the average flying ability by factors greater than 100, meantime their watches are probably not set to within 2 minutes. Nobody ever expects to be told the time to an accuracy of greater than a minute, and time remains the keystone in navigation.