It's American walnut and/or Claro (?) with huge pores. I am by no means a finish expert but have finished my share over six decades of putzing. I have been viewing this many times since posted. All I have ever seen posted about slacum finishes has been related to English walnut and associated subspecies. If you are married to linseed oil finishes follow Keith's advice. An easier and less labor-intensive finish can be achieved with Daly's products. Seafin sealer and Teak oil finish. Sealer x 3 coats to seal the pores; sealer mixed with rottenstone (make mud) and wiped between coats across the grain tilled filled, sand back to desired grit 400-600- finer again is reserved for fine woods; finish with Teak oil. Durable, forgiving, looks great and easy to repair. While I do not work much with AW, I recently finished a stock set for an 1886 Browning project with some similar issues. Thin shell walnut is your friend. A quick perusal of fine furniture finishing techniques would likely be beneficial as well.
