You might try scrubbing it with a dilute solution of white vinegar.

I just last Friday had both my walls and floors (I have the same large format tile on the walls as well as the floor) grouted with Mapei Warm Gray. It's gorgeous, I love it on the walls, but somehow I'm finding it a little to light for the floors, even though I like and wanted a uniform look. Don't get me wrong, I'm very happy with how the bathroom is turning out, but I think I'd have gone a little darker on the floors if I had it to do over.

The hardwoods are extremely valuable. I would work in a medium brown that eliminates all yellow/orange/red. To do that, the stain will have to have a green undertone. That means a specialized mix for your situation. A HIGH END hardwood flooring professional can do this for you. It will take a couple of weeks to find the right colour/mix. The professional will come in and strip a section of flooring (like 20ft x 20ft). S/he will then apply different stain mixes to this area for you to view. You will then choose the colour that is the best fit for your situation. At a mutually agreeable time, the professional will come in and refinish the entire house - including the stairs (quite expensive but well worth it). The banister can then be stained/painted to match the hardwoods. The only time I would look at moving away from hardwood is in the DEEP south or in very hot climates. And it would be VERY helpful if the entire house was built on concrete and not wood joists. Tile is very heavy. It often requires wooden subfloors to be beefed up (decrease the deflection rating = no movement that would damage the tile). The wood is worth saving - unless you absolutely hate it (disregard the colour). If you love tile and wish to have it everywhere, then go for it. But realize that hot climates are some of the few places where this is considered normal. Everywhere else (any place with "winter") would find this a nuisance should you wish to sell the home. Besides, you mentioned budget. Ripping out the hardwood and replacing with tile would be double the cost of refinishing the hardwoods.

Looking at their grout chart, Stone Beach or Sugar Cookie. Are you sure your choices are Ardex? The color names do not seem to match the colors that they offer? https://cdn.ardexamericas.com/wp-content/uploads/2018/09/ARDEX-TSIS-Grout-Color-Chart-Brochure-download.pdf