I agree with you fully. The parent comment's focus on the use of arms during a climb seems rather silly.
Downshift, focus on maintaining consistent, sustainable pedal pressure and cadence. Ignore road speed entirely. If you're doing it right, even the steepest hill is just a long stretch of flat road.
Focusing on the role of the arms in applying force to the pedals only makes sense if we are talking about single speed bikes, fixed gear bikes, or racing. It does not make sense when we are talking about trekking.