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I have a 6 speed and on my way home from town there is a n 18% hill. Sometimes I ride up it, sometimes I push. It depends how much crap I have in the bag vs how hard a day I’ve had. There is never any shame attached to pushing a bike up a hill ever.
As for locking it up or taking it in to shops? The nice people in John Lewis Norwich have popped it under the till point (locked folded) for me before and I’ve spent a few hours around the city and come back later to pick it up.
Bear in mind in those pictures it has an 8 speed alfine hub gear on the back, but the gear range isn’t far off a 6 speed wide range with reduced gearing at the low end but yes imagine it’d be more than fine (legs on the other hand…)