I have come across the chipping paint issue before. I find a piece and it is lovely but the original paint job is chipping off everywhere or really goopy and in awful condition – yikes! This is no fun to deal with and honestly, I pass up on many of these pieces because of the work entailed.
This was the case with the Corinth Blue Dresser. 
I sanded down the goopy paint and in those areas, used the layers of previous paint colors to create an interesting and pretty look. I used General Finishes milk paint in Corinth Blue on that piece and it was a lot of fun to play with.
Presently, I am working on Lily’s Armoire. The whole piece is covered in chipping paint. Chipping paint is fine but honestly, I like controlled chipping paint and this piece is seriously not controlled! It is flaking off everywhere. 
I took my orbital sander today and went over the areas (door, drawers, etc.) that were the worse for wear. I then used my Elmer’s Wood Putty and applied it over the bumps. I let that dry and went over it one more time with the sander.
I will add that with very old pieces, one needs to be careful with the old paint. It could be lead based and dangerous when inhaled. I use a mask when I sand my pieces and I do it outside in my (super duper hot) garage. 
It is now smooth to the touch. After one coat of Annie Sloan chalk paint, this is where it stands. The paint looks very uneven. 
I will sand it again over the weekend and apply another coat. This technique may need to be done a few more times for a good finish. Another option? Play with it. I may add a lighter coat of gray on top and sand it back and see what happens. I’ll keep you posted!
I am so glad to hear that someone else wants to “fix” chipping paint. I worry that it will flake off everywhere. With your talent both projects will be amazing. Cyn