Here is the technique for removing plugs without damaging surrounding
wood:

Since your material is 9/16ths thick my guess that the plugs are
either 3/8th or possibly 1/2" dia. and they have to be rather shallow.
I think I would try to drive a No. 8 screw right into the center of
each plug. This should force the plug up and out without damaging the
bung hole.

However, on reviewing everything you have written I wonder it if would
not be more cost efficient to just rip everything out and install a
totally new cabin sole. If you wanted to save money you could think
about using teak/holly plywood which goes for around $100 for a 4x8
sheet and is available in 1/4, 3/8, 1/2 and 3/4", perhaps even
thicker. It real easy to lay. If you use FastCo two-part, one to one
mixture epoxy (which comes about the consistence of peanut butter) you
can spread it on the plywood with a notched spreader, lay it down and
weigh it with bags of sand. This way you don't even need to use
fasteners.

If you want to go the solid wood route you should be able to buy 1/2"
vertical grain (you should use VG for a deck or cabin sole) teak for
around $10 to $15 a sq. ft in 1/2". Holly is hard to come by but no
one will know the diffrerence if you use hard maple instead. You
should be able to buy that for around $5.00 a bd. ft.

Total up your cost in materials and you will see that if you end up
doing it your way (trying to reuse your materials) you are going to
end up working for about .50 cents an hour. Isn't your time worth
more than that?

Good luck

Joe