If the chests are the same size you can lay the door on top with each chest at the ends. I would then attach with screws on all corners.
I would think that a hollow core door would sag in time. Perhaps a sheet of plywood with laminent coat. It is sold ready made in several sizes.It will last for decades.
If the hollow core door is not more than 18" wide, you should be fine. I used one (across 2 bookshelf units) 25 years ago for a sewing desk. It has both the sewing machine and the serger on it. I use them frequently and have never had any issue - no sagging, no bouncing when sewing. I did put a nice strip of wood around the edges, primarily for looks.
How will you deal with the door knob hole? Will you use the entire 6ft8inch length? If you cut down you have to rebuild the edge. The door idea doesn't sound good.
Yes, the door will sag with the weight of a sewing machine with a span over 42”. Plywood will sag too with a span over 42”.
The doors shouldn't be cut without putting blocking in them after the cut.