A chelate is something which grabs (‘complexes’) heavy metals (most of all iron) in a tight grip so they can no longer redox cycle and oxidise unsaturated fatty acids (i.e. make them go rancid faster, a process deliberately initiated by addition of metal salts to speed up drying of linseed oil paints and varnishes). Now, your toothpaste is at pH 9 which would be a good pH for citric acid chelation and contains a lot of bicarbonate. At that pH and carbonate content, most of the heavy metals are in insoluble carbonate, oxide, and hydroxide form and don’t do much other than just hanging around. Adding citrate (assuming there is enough water to actually keep it in solution) would be catastrophic (in theory) because it will make said metals become soluble and bioavailable for micro-organisms. From a chemical and microbiological point of view, adding citrate to a toothpaste is an utterly bad idea.
What citrate does is to chelate calcium. In case you use hard tap water for cleaning your teeth, it will make it feel softer and, should you have soap inside your toothpaste, also boost foam and produce a nicer lather. This is likely the only good reason why you should add citrate (for example as sodium citrate).
In anhydrous products, you’re well advised to add an oil soluble antioxidant (vitamin E) to protect the oils from getting rancid too fast. You can’t do much with a chelate (because they require water) in this regard (at least not with ‘natural’ and potentially edible ingredients) except adding a tannin rich plant extract such as rhatany which is great for the gums but also bears the chance that it precipitates iron ions (however, having an anhydrous formulation might forestall the latter effect).
How to use sodium citrate in oil: as a suspension exactly like you use sodium bicarbonate.
BTW I wouldn’t bother too much about HUMANS drowning in plastic but rather the whole rest of our planet
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