You definitely have to remove all of the paint and especially all of the powder. Any paint that you apply can only adhere to the floor if the surface is completely clean. You mentioned that your house is about 60 years old. If the paint that is turning into a powdery substance was applied long ago, there is a good possibility that it is lead-based. If that is the case, you should stay away from the powder. It is very dangerous, particularly for children and pregnant women. Professionals can clean the floor. If you do it yourself, wear full protective gear to remove the excess powder and then build a subfloor. It is much safer than digging and scraping away at the existing paint. It will be a little bit more costly, but there are advantages. Cover the floor with carpeting, put down hardwood floors, or paint the plywood. You will have more options and it will be much safer.