I'm a General Contractor who hired a flooring subcontractor to repair/refinish 5/16" x 2-1/4" Oak strip flooring. Conventional parallel strip flooring in halls and bed rooms and then approximately 12" x 12" parquet squares of the same material in the kitchen/dining room. There were areas where high traffic and/or moisture exposed the face nails. Around each nail is a black spot. Now that it is refinished it stands out and looks really bad. This Sub came recommended by the Owner and I don't want to jump on him unless I know for sure what he could have done. If I discovered a flaw like this on wood work where I was working I would have pointed it out and offered solutions before applying the finish. Make the Client decide what to do next. Searching on line I find recommendations for using hydrogen peroxide or oxalic acid solution on the bare wood after sanding to remove the stains. Is this proper practice? Was he just being lazy or not wanting to take the extra time? Same Sub did a few other things wrong but those I'm certain enough as to proper practice to just ask them to be fixed.

Exposed top-nails need to get set below the surface. The steel will rust from water used to clean the floor or from a spill that does not get wiped up. The black spot does not sand out if the floor has been damp for any length of time.
When I sand a floor that has rust spots I take the time to set the nails with a larger nail set or use a chisel to gouge out a little of the wood surrounding the nail that has become blackened by the iron salt after setting the nail below the surface. The matching wood patch makes the hole disappear after sanding. Bleaching with oxalic acid would be an extra step that I would not recommend.
This black spot really shows up on a natural finish which will yellow with age. The carpet tack strip can also leave black spots where the ring shank nails hold the tack strip down. The wetness that comes from cleaning before the holidays in the fall or winter when the floor is cold allows the rust to happen. Time to use a quarter inch chisel, then fill the holes with the matching wood patch which blends with the wood. The client usually does not even know that the work was done.
Once the floor is completely finished it is a problem to get the black spots to blend in after the gouging and filling. Colored putty could be used but ought to be covered with another coat of finish, extra work.
Some contractors charge extra just to set the nails! I've seen rows of nails exposed after sanding and finishing the floor which is not up to the standards of the industry.

The reason that I wondered about the oxalic acid or hydrogen peroxide is that the flooring was only 5/16" thick when new. House was built in the middle 60's and I don't know how many times the floors have been sanded. Is there a minimum thickness of material where the nails could still be reset? Was thinking that after the current sanding if you could get rid of the stains and then cover with a good layer of finish yes you'd see sanded nail heads but your eye would not be drawn to those black spots.

It takes a lot of sanding to sand enough that the nails can not be set and filled. If the flooring is 3/16th inch thick then the nails could be set almost an eighth inch, if they were showing. The little brads hold the wood not only with the heads of the brad, but since the brad has a diamond point that spreads the fibers, the spreading of the fibers press each board together so they hold together.
Do almost all the nails show? Is fastened down well around the black nail heads? I have sanded floors where some of the boards are hardly more than an eighth inch thick where the sub-floor was not flat to begin with.The wood can get warped from the moisture that it takes to make rust stains. This may cause extra sanding.
Exposed nail heads will be susceptible to the finish wearing off and rusting again.
Try setting the nails a little and fill the chipped out area where the rust is with putty.
It is much easier to do this before the final sanding cause the filler will color the nail head even if it is very thin. Putty is the last resort, but ought to be done before the final coat of finish. Shallow putty will smear while buffing between coats.