Silica is not even close to hard enough to polish sapphire, and most toothpaste, instead of containing silica, contains chalk, which is not even close to as hard as silica. The problem with brushing your teeth with silica is that silica is not a very mild polishing agent; it's a pretty aggressive one, and it will wear away your tooth enamel pretty quickly. If those are your permanent teeth, that enamel ain’t coming back.

Silica and maybe even chalk will work fine to polish glass, though.

I have a correction. I went out and bought a tube of Colgate ("Triple Acción! Extra Blancura!") and it does indeed contain silica as an abrasive. (And Colgate is hardly some weird off-brand toothpaste.). The difference is that it’s hydrated silica, which is amorphous rather than crystalline, and is somewhat softer than normal silica. (And, as an extra special bonus, not carcinogenic like normal silica!) This is the same stuff known as "silica gel" that comes in little pouches with your electronics labeled "DESICCANT — DO NOT EAT".

Amorphous hydrated silica is still hard enough to polish glass and abrade your tooth enamel. Chalk is not, although I'm not sure it's hard enough to abrade away your tartar deposits.