I think you've done well so far. The repair looks solid and should last.
Since you've removed some of the old varnish, I'd remove all of it. Some sort of a stain would help hide the repaired cracks, but common stains of the Home Depot/Lowe's variety tend to be cloudy and obscure grain if that's a factor. Birchwood Casey has a walnut stain that's a nice brown with red tones. It's water based, but can be cut with denatured alcohol. Laurel Mountain Forge has a range of stains that can be mixed to provide a wide range of colors. They're alcohol based, too. If it looks too dark, wet a rag with denatured alcohol and wipe it off.
I don't know what Enfield stocks were finished with when they were manufactured. Personally, I refuse to use boiled linseed oil (BLO). It never fully dries and provides zero protection against moisture. Formby's Tung Oil works great and dries. There are other finishes out there, but tung oil looks good and is easy to apply. Let us know what you decide to use.