Stain colors wood---how are you going to make sure the darker stain staya only on the putty and does not get on the wood?

Sounds like the nails were hand driven with a hammer---that leaves a crater around most nails. That is acceptable since the results are not intended to be seen---that is why it is called rough framing.

Manufacturers claim many things for products---stainable putty is not one I have seen that works even sorta well. The problem is not so much the fact that putty by its design is very dense, it is because every piece of wood has a different porosity and takes stain differently.

Now, what to do for your stair treads. You can try a lot of things. And you might find something that works. My suggestion is to either primer and paint them (with several coats of polyurethane as a protective finish)----replace them with real' stair treads intended to be used as finished treads----or cover the treads with 1/4" thick covers (You could use plywood and hide the front/side edges with metal stair nose cap).

If you replace the treads, use countersunk screws and wooden plugs to cover the screw heads.

If you use the plywood idea, use adhesive---applied with a notched trowel ala tile----and either finish nails or finish screws. Then stain the wood(only sand to 150 grit) and use a product called Stain Putty to fill the holes. Stain Putty is a powder that is mixed with the stain used on the wood to make an exact color match filler that is applied after the stain process. Sand smooth(now use the 220 grit) by hand and finish.

I bet the putty you used was not waterbased. Only waterbased wood floor filler will do an acceptable job, as it does not seal the surface in any way. Also, as you have learned, filling holes is best done after staining or sealing, at least that has been my practice with wood flooring.

You can try and color the pale spots with a darker stain shade, as you say, or attempt to color the spots with an acrylic craft paint mixture of a color that will blend with the surrounding stained surface.

Either is worth a try.

If the nails were driven with a hammer the wood around the nail was crushed. Wood stain works by lodging pigment in the wood fiber. Wood that's been smacked with a hammer will hardly take stain. And stain, after it's dried, won't really take more stain darker or not.

You could try sanding back one or two pale spots, as an experiment, and restain with your original color. Sand the spots with 100 grit, leaving the wood coarse, and restain the spots. They'll be darker and maybe dark enough. If it works do them all and sand the rough spots out after your first finish coat dries.