Other types of tooth fractures can and do actually heal, via natural remineralization, some toothpastes are designed to help do that.
I have observed that keeping the broken cusp in it's proper position, in other words never moving it, seems to stop infection and bleeding ( as I described in a reply above ).
So there is a chance that the string ( or something ) idea I typed above could have some sort of percentage chance of being successful.
I think that probably the reason dentists insist on extracting it is to not run any risk at all of patients health being affected by infection, regardless of it being an idea that could have some sort of percentage chance of being successful.
Since I can find nothing on the internet of people doing this string ( or something ) idea DIY, I will have to have it extracted.