The inspection itself is normally performed by the housing authority at no charge to you. Your manager is right about that. The money goes to repairs and to the days the unit sits empty.
The items you listed are the usual suspects. Rough figures, and these swing a lot by market: battery smoke detectors run about $30 each and carbon monoxide detectors about $50, so a two-story duplex unit might be $150 in detectors. A GFCI outlet swap near a kitchen or bath sink is often $150 to $250 per location with an electrician. A stair handrail is commonly $200 to $400 installed. Window locks and sash repairs are small, $50 or so each, but they add up across a whole unit. A missing or non-working permanent heat source is the expensive one, because that's a furnace, not a fix.
On a 1961 building, paint matters. Housing built before 1978 falls under lead-based paint rules, and chipping or peeling surfaces will fail. Depending on the surface and the rule that applies, the repair may need someone trained in lead-safe work practices, which costs more than a painter. Those requirements have both federal and state layers, so ask the authority which standard they inspect to.
The cost your five-week landlord actually paid was vacancy. Scheduling the first inspection takes days to weeks, and a fail means a re-inspection on the same queue. Two rounds can put you six weeks out from a signed contract with no rent arriving. Fix everything on the first pass, even the items you think are borderline, because a second trip is the expensive item on this list.