How rare is it to die from dental infection?

I've had a ton of dental procedures. So, I consider myself a quasi expert on what u can and cant get away with. So, first, the pain is caused by infection which serves as your body's alarm to get something handled. Infection is caused by an overgrowth of bacteria which leads to tooth decay which weakens the structure of your tooth. Those puss pockets and swelling ur describing occur at the side(usually on either both or one side of the gum line where the tooth is). You do not have that because you took your antibiotics and killed those bad guy bacterium, therefore you feel no pain because you dont have infection...right now. Now, your fracture allows bacteria(which we have naturally in our mouths) and acids access to the nerve, so when the number of bacteria get to the crucial point again, then u will have pain, swelling, etc. You also run the risk of having your tooth fracture vertically to the root, in which case you will not be able to get another root canal to clean out the bacteria nor crown to save the tooth, you would need the tooth extracted and a dental implant put in...much more expensive and longer process not generally done same day, but over the course of months. Meanwhile, keep in habit of rinsing your mouth with salt water and spitting once or twice a day to keep bacteria at bay and be sure to brush your tongue when you brush your teeth and avoid chewing chips or hard crunchy things period or if possible just not on the side where the fractured tooth is. In your case because the tooth is fractured pay attention for pain when eating and eapecially check your mouth for a abscess which looks like a lump on your gum.

It is just quite possible.

It is possible.  I had an infected tooth and could not get into the dentist for 5 months because of the Pandemic.  The tooth was removed a week ago and I have been on Antibiotics for a week.  The pain stopped within 2 days and I actually have some energy the last three days. 

Yes, it's possible, but pretty rare. The bigger risk is developing endocarditis or septic arthritis - both are common when you have a long-standing infection in the bloodstream.

the infection in your mouth isn't what should be bothering you it's the huge gigantic case of hypochondriac trollitis you have

Dental infections are very close to your brain, which makes them far more serious than an infection in one's toe, for example. If you're willing to assume the risk for not getting treatment, don't complain if you contract bacterial meningitis.