Is a root cellar supposed to be wet inside? No damp/waterproofing, spray foam joints/air intake, no drain tile. I suppose direct and honest comments like mine is what kept the OP away for 10 years.

That your existing concrete is in good shape is key. Any crack in it would transfer through to the new material. So long as you create proper finish levels, you should be in good shape. Stamped concrete can look terrific. It's biggest drawback is the color not being integral to the slab, but just surface treatment. This is really no problem unless it is subject to harsh wear, which in this instance is probably not going to be the case. If the surface gets a chip or gouge, you'll see near-white concrete below. In most residential situations, it doesn't get that harsh of treatment.

Asking a contractor to match concrete is akin to calling in a baker and asking them to match the wedding cake, that you happen to have a sample of in your freezer from a couple of years ago. While they can get it close enough that everyone will agree that it's white and cake, there are going to be differences that the more discerning palates will undoubtedly notice. In your last picture, it looks to me like the contractor used a coarse broom finish, which is an extremely common finish. How rough it is depends a lot on the broom and the broom operator. There is no way to grind/"sand" this smooth without changing the character to yet another version of concrete that doesn't match your steps. If your contractor thinks there is, I suggest you'll be in for another surprise. (You had best get a sample of the end result before committing to the change.) Myself, I much prefer a ground finish over a broom finish. It's smooth and starts looking like low contrast terrazzo. (BTW, do not use an aggressive diamond blade for grinding or you'll be grinding it a second time in order to remove deep grooves. Grinding with carborundum stones would be all you need for smoothing. Also, keep in mind that it's not a 1-hour job. It would be more like a day of grinding ... so would cost extra.)

Hello Kyle! The better option is painting the concrete. Using the concrete stain will look bad, in my opinion. In the picture, I see that area under your concrete ladder is Has an uneven surface that may be bad for water balance when raining. Anyway, I saw that you want to do it by yourself. Contacting a concrete company like this one https://www.fremontconcretepros.com/ is not a bad idea. But in my experience, it is better to pay a bit to professionals than having future problems that may cost you more than addressing concrete companies.