You're asking for the impossible. What would being 'natural' add?
Skin safe alum isn't natural either.
Deodorants containing baking soda work perfectly fine for a lot of people. That it does not for you, doesn't make it a product that 'shouldn't be used' in general.
Search Etsy to see what works for other people, or visit the swiftcrafymonkey blog and search for deodorant. There's a wealth of information over there.
I agree with all that, but while the original poster may have been exaggerating to say baking soda "shouldn't be used", it is kind of on the edge of tolerable. Arm & Hammer had a stick deodorant that advertised its baking soda content, but that was actually a gimmick because there wasn't enough in there for that to be the basis for its effectiveness--in fact it was probably on the acid side overall.
There are many underarm deodorants that rely in part on affecting the skin's pH. The recipes using baking soda do so to neutralize volatile acids responsible for some of BO, keeping them in a salt form that doesn't get into the air. However, skin doesn't like to be alkaline. I suspect that some of the home recipes containing baking soda similarly to the Arm & Hammer products don't have enough of it to be responsible for their deodorant effect. Other deodorants are acids; the aluminum-based antiperspirants (which are more effective as deodorants than they are as antiperspirants, reducing odor far more than they reduce sweat production) are more effective, but also more irritating, the more acid they are, and as with baking soda, different people's skin will draw the line in different places as to how much they'll tolerate.
However, to some extent any organic acid will be somewhat antimicrobial and help to slow down axillary sweat's spoilage. This property was taken advantage of in a roll-on and stick product called Fun 'n' Fresh, which was a minimalist underarm deodorant pitched for use on children but at least somewhat effective on adults. Its active ingredient was triethyl citrate. This was an oil-soluble ester that would slowly hydrolyze to release citric acid. The slow release meant there was never enough acidity to be irritating, but the acid production would be maintained for many hours. I'd like to see another product come out with that principle now that Fun 'n' Fresh is gone. You can rub your armpits with lemon for its citric and other acids, but it might sting pretty quickly and won't be effective for long.
Various essential oils have long been used in underarm deodorants for their antimicrobial properties and also their scent. Mum has been a commercial version of such a product going back to the 19th Century. However, once again people will differ to the degree they'll tolerate what is in effect an antiseptic on their skin. That's why synthetic antibacterials were developed for such products, not to be more effective (usually they're not) but more tolerable. The antimicrobial essential oils usually contain phenols, benzyl alcohols, or benzoates; a classic example would be thymol, the phenol which is in the essential oil of thyme.
