Wood – Dirty water stain on oak worktop

I'm assuming this was a standard oil finish like a Danish oil or tung oil. They will harden. If the oil finish was a mineral oil, commonly used for butcher block tops for example, that oil will not harden. But even so, one coat of oil finish is not that much of a finish.

You should be able to apply stain to it. In fact, a hardened oil finish will help that stain so it is applied fairly evenly, although it may not soak in as deeply as it might otherwise. There are different kinds of stain, and that will be a factor. Thus there are dye based stains, which get into the wood. With an oil finish, the dye will not penetrate as well. Pigmented stains (essentially thinned paints) have pigment in them. (Ok, maybe that is obvious.) They are designed to sit on the surface of the wood anyway.

As far as a varnish finish, yes, a hardened oil finish that has soaked into the wood will not do anything to stop a varnish from hardening. However, if the oil finish is a non-hardening mineral oil, then varnish will probably not adhere terribly well to the wood. So really it matters what kind of oil finish was applied, and what is the purpose of this top.

Sealers are important for pine because the stain will tend to seep in otherwise, making it look blotchy.

Sealing after the fact is not optimal but probably better than not sealing at all. I would recommend re-sanding it before applying the sealer.