I trim out houses, lots of nail holes to fill. I don't use any nails on cabinets and furniture that will show. Most I know in the business use a product called Color Putty, may be similar to Minwax, comes in a clear plastic jar. It's soft and two or more colors can be kneaded together for any color match if you don't like the stock colors. The work is stained, top coated and installed, putty placed in the holes, then the excess putty buffs off with a clean cloth. A final top coat is applied to the trim. It must be used in this way but it's very easy. Today, it's common to stain and apply one coat of varnish to the moldings before they are installed. This is much easier than having to do all that work after things are in place. No dirty finger marks to remove, no cutting in against walls, etc.
Any filler that is stainable has to be used before staining and top coating. Excess has to be sanded. You can't just sand that small area or it will show when the stain is applied, the whole piece has to be sanded for consistency. Very labor intensive. It also means, with house trim, that the moldings get installed in the raw. The edges of the filled nail holes accept stain differently than face grain and the filler. This makes the nail holes more pronounced, visually. There are a few that still do it this old fashioned way but the majority do it as I described above. Traditional fillers often get pushed out of the wood over time and stand proud of the surface. Wood will shrink and swell with humidity changes over the years. Doesn't seem to be a problem with the Color Putty.