Do the sanding, collect the sanding particles, add stain, mix with glue, fill holes.

Do some testing, the sanding particles may stain differently than the wood. You may need to adjust the stain amount.

OH, duh!
Just shows to go you when you get something in your head, how you can miss obvious alternatives.

Thanks! :)

>Sand it smooth and you'll never see it.

This has never been my experience. First the glue, about any glue, sizes the wood and prevents penetrating stains from, well, penetrating. Second, sanding dust is essentially random, meaning a significant portion of end grain, that usually takes stain quite differently.

Likewise, "stainable wood putty" does take stain, it just doesn't always take it the same as solid wood, for one reason, different pieces and species of wood take the same stain differently. Heck, even different parts of the same board can take stain differently (think oak pores, maple grain, etc.)

Another option is to stain, put on all but the last coat of finish. Fill the holes with two or three closely matching fillers (there are a bunch from soft waxes, putties, to resin sticks). The different colors will allow you do faux in some grain and detail. Then apply your last coat of finish to seal and match the sheen.

Here is a link that might be useful: Here's one filler option for the DIYer