How to repair cement/flowable-fill slab cracks
I mis-interpreted your picture.  So what we are seeing is a level concrete floor with carpet tack strips in the foreground area.  But I'm not fully understanding the foreground of the picture you provided.
It appears that we can see the edge of the concrete pad in the middle of the floor.



If you had no issue with the floor with carpet in the past, I would just leave it as it is if you are planning to recarpet.  
The thickness would indicate it is cement. Repairing the bad spot like under your foot will not stop all the other fine cracks from continuing to crack further. It took 50 years to reach this point. If you took it out, you would almost have to do another concrete pour to bring it up to the level of the other rooms, i.e., bath rooms to the same level. Otherwise you would have to tear it out for all of the second floor to bring them all to the same level. 
Unless you have some serious concerns, I would leave it as is. 
Will you still be there in 10-15 years?

If repairs/sealing of the cracks is to be done, I would go with two types of products, liquid/self leveling for the thinner-fine cracks and a heavier product for the larger cracks.  For the liquid, look at products similar to Dap 37584 product and for the larger cracks, Dap Ready-Mixed Patch.