Accept it for what it is: real cherry. Cherry will darken quickly, and doesn't oxidise worth a darn if you are after a rustic brown. It's the wrong wood for that.

Your first step is to complete the stripping and sanding to get the whole chest into the same state. You have to do that or the pieces will never match.

THEN ... use a wipe-on oil-based brownish stain that has a slightly green tone to it (probably one of the named walnut ones). If you are ambitious, you can use a dye with a green tone first, to cancel the reddish tone, then a brown stain ... but it's tricky.

This will tend to cancel out some of the red, but not hide the grain.

Then finish it with a wipe-on satin topcoat to protect the wood. I use General Finishes from Rockler.

First, give it a rest. No one cares if you paint or stain your piece purple. It's the how-to that counts.

Brown will increase the red some, as lazygardens mentioned above. So a green tint is the way to go. Talk to a woodworking shop, not HD or Lowe's. Definitely do a light coat and repeat to get the depth you want.

I was experimenting with cherry and oxidizing it does work but not as you'd prefer. Plain vinegar did nothing. Vinegar and the steel wool turned it immediately ebony. No in the middle there.

As far as a matte finish, there are a few things you can do and they're not wax.

I use the base used for the darkest color paint, flat, exterior latex. Just the base. Goes on white and dries so matte you can't tell where you put it on. I am enamored with this finish and am using it on a bunch of antique doors I've stripped and some I let get very chippy in the yard for a year or so. It's very inexpensive.

Duraclear also makes a wonderful, stupidly expensive matte varnish. The same result as the paint base.

There are some products by Beautiful that dry matte, but it's a multiple step process and to me, not worth the time.

Oh - almost forgot.
I stripped some antique cherry cabinets and also wanted to get the red out. After the initial stripping with Citristrip, I did a 2nd coat and let it sit overnight (plastic bags pressed into it) it sucked the stain right out of the wood getting rid of much of the red. I wanted more out, so I literally bleached them multiple times with Clorox. I could use oxcylic acid, but I was doing it inside, so not. But Citristrip is absolutely the way to go with stripping, and its stain removal properties is a serious bonus.

Just keep playing with it. Know that stains or dyes put on freshly sanded wood are virtually impossible to get out, so experiment on the back of the trim or something.

Plain vinegar did nothing. Vinegar and the steel wool turned it immediately ebony. No in the middle there.

High tannin content ... if you dilute the solution you can get shades of grey.

But it takes a lot of testing.

That's what I figured. I've had to use strong tea and let it soak in to ... tanninize(?) other wood, so that made sense.

Not worth my time and really just an experiment. I'm into the chocolate brown and cream they became when I bleached them. Another experiment. ;)

I would try a gel stain also, as that is more like a paint.