If you are going to use the press I would not stain the wood. Marine Varnish might be okay, I doubt anything else would hold up.
If all the wood is being replaced it would be hard to find a wood more suitable than white oak. You could leave it unfinished and it would last longer than almost anything else. It is great for use where it will get wet and it's very strong. Most all wine barrels have been made for it for more than a century.

X 2

The main body is red but the gears are black. so about 50-50 on what you can see.

If you stay with cherry, try to use the straightest grained pieces you can and trim all the lighter sapwood away. The heart wood is pretty decay resistant and looks better to boot. and like most woods it will get much darker with age-plus the grape juice-I have no idea what it will do to the color.

Of the 3 cherry. It brings out the grain. Are you staining the outside, inside or both?
Me personally I'd probably let the grapes stain it throughout time. If you're staining just the outside Cherry would be my number 1 then possibly cab but not as rich and deep. Wipe off right away so the stain isn't as dark

Ok. Well if your bent on just those 3 choices, I stick with chestnut and then some kind of food grade clear. Like they use on butcher block. Plus the darker you go the less any staining would show.

If it were mine, I would use the cherry stain, but I would give it another coat. Cherry wood will absorb the extra stain, and it will bring out the natural grain even more than what your pic shows, without making it significantly darker.
I dig cherry due to the character commonly found in the grain. Your other 2 choices will diminish the appearance of that trait, I am afraid.