To the OP. Firstly fettle the inletting before doing any stock finishing. I like the CS stocks but they all benefit from some sanding to get them to fit your action properly.

Danish oil is a type of wiping varnish. So not really a true oil finish, it's a mix of oil, thinners and varnish. It doesn't penetrate very far, but produces a durable finish with minimal effort, you can do a "coat" in about 10 minutes and repeat once a day until you're happy with it. Then leave for at least a week to fully harden. The end result doesn't have the feel of a true oil finish, unsurprisingly it feels like what it is... a variation on varnish.
You can cheat a bit and once the DO is hardened, lightly rub with synthetic wire wool, 2000+ grit paper is better, and then use a clear wax on top.

A proper oil finish will enhance the grain significantly more, but requires much more effort. Maybe 15 hours work to get a good result, again needs leaving once done, a month is about right.

As for supermarket oil being a drying oil, or any other for that matter, there's an easy way to check. Put a couple of drops on an impermeable surface, glass is perfect. Leave it overnight, if it's still gelatinous the next day then it's not a drying oil, if it's become plastic like then it is.

CS500 oil finish.