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It's difficult to get under a nail head unless it can be moved even a small amount proud by prying up under the skin. If it could be done with minimal bending then it may be possible to work a putty knife under enough to get them out. I made a nail puller out of a putty knife to get the ones out on my 29 Sport Coupe and it worked fairly well. Some of the heads came off during the process but they were rusty and the wood was shot. The shallow wedge shape with a narrow slot works best. I had to smooth out corners so it wouldn't tend to dig into the sheet metal while tapping on the end of the knife. I took the hand grips off due to beating them to death. It gave more clearance to get closer down to the metal as well.
Having removed so many rivets in my day by drilling the head. A person could use a small drill but the nails may be pretty small so not an easy job either way. If the wood is salvageable then getting nail remains out would also be a job.