What should I do with this cedar chest?

Personally i would just leave it original. It has some history. The tag is part of it's provenance. Most of the time they are missing. Refinishing can bring the value down by 20%. I would say the value is around $300.

https://www.hunker.com/13412447/determining-the-value-of-a-lane-cedar-chest

I would frame that cool note! I don't know about the value other than to you. With some stripping and sanding, I bet there is some nice wood under the worn finish. Personally, I would clean it, strip the finish, sand it smooth and use a gorgeous deep wood stain and clear sealer. If you want a paint look, whitewashing it would be gorgeous, too.

I would leave it as is. If the top bothers you, lay a quilt or blanket on it to hide the wear and tear. You're right that it will devalue it if you do anything but keep it clean.

I understand your wanting to refinish the chest. I have one that is in bad shape also. I keep it covered with a quilt I made just because I cannot afford to have it refinished and I am not confident enough to do it myself. I really need to ask my children which one of them is going to want this when i pass and if they want it in the original but damaged finish or if they want it refinished or some other finish.

I inherited from my mother and it was not in great shape but has survived my three children, multiple moves as well as the same with mom.

I think if you intend ...to sell it leave as is. If you want to display it in your home but want to make it look better go ahead and refinish it. It's totally up to what works for you.

I would sand it down then paint it. You will be surprised at how beautiful it can be. I'd use a chalky paint which doesn't need much prep work aside from a light sanding to remove that old varnish. When you are finished painting I would seal it to protect the finish.