I would try staining it darker. You can use a sponge to apply the stain. You can get paintbrushes for stain from Lowes but they are a bit more expensive. I'd get painters tape. I know you said you some want to do a lot of prep, but it's going to take prep whether you do paint or stain.

This is going to take some preparation. Such as:

• Putting Painters tape around the trim.
• Putting a tape down or not. it's optional but recommended.

You could paint all or most of it

I know it seems daunting but if it is in good condition and you like the style I would suggest painting it. Start with one room at a time. Take your time and don’t think about how much there is and how long it will take. Just start with that bathroom and see how satisfying it will be when you are done. I wouldn’t stain it. Darker brown is still brown and there will just be a whole lot of dark brown instead!

if you use the wrong paint over varnished wood, it will partially peel off. a terrible project to fix down the line. that's what the previous homeowners did to our house and I curse them all the time for it.

you'll need to clean and prime it, and choose the appropriate type of paint that goes over whatever that trim was varnished with. (go to a real paint store and ask for their advice.)

for what it's worth, I love wood trim in a midcentury. it's not dated for that style of home. but if you hate it, I understand.

There is no reason that the doors and the trim in the bath could not be different from the rest of the house. I would maintain the look of the current doors and trim in most of the house.