Both birch and maple can look good with dark stains, but they don't always look good because they can blotch (absorb stain differently on the same piece of wood). Ask for samples - not small pieces, but entire doors. The final result really depends on the process used in the factory. Birch and maple are more likely to result in a more neutral dark brown, while with cherry it almost always retains more warmth/redness.

If you want a dark stain with no pricing upcharge, you should look at oak. Perhaps alder if it’s offered. Birch and maple both are best as light to medium stains. Both tend to unavoidably blotch when trying to go darker, even in the hands of professionals. It’s just how the wood behaves. You can combat that with a tinted trsnslucent top coat, but that masks the graining. Cherry has a beautiful grain and takes darker stains well. It does have a price upcharge, but not as much as something like walnut.

Here is a darker stain on oak for a showroom that I designed. You really do not see grain when it’s dark.

Here is a slightly lighter stain on alder in that same showroom.

Here is a cherry with a medium stain in the same showroom. It’s a much prettier grain up close than he above two.

@mlb418, Yes, Alder will ding and scratch more easily than Oak. Look up the Janka scale ratings for the woods you're interested in to see the exact specifications on hardness. Cherry is softer than Oak but harder than Alder. The softness of Alder is one reason why so many Alder cabinets are offered in a "rustic" or "character" style with imperfections already included in the new product. However, you'll see rustic Alder in expensive mansions as it is a popular wood trend at the moment.

You have to get samples. Bertch offers eucalyptus and hickory which may be less. Quarter-sawn oak doesn't show open grain in medium dark stain colors. Flat cut oak has open grain that turns black and become prominent when stained. Cherry can turn dark if sunlight hits it every day.