bugs have a way of finding stressed plants .... and a tree that caught fire is probably stressed .... 3 years isnt much time in tree time.. which is counted in decades ... probably one of the wooly aphid type creatures .. but i cant tell from the pix ... but it looks awful close to the house... it might be worth your while to have a certified arborist give you an onsite inspection of the trees overall health ... add your big city name to the link below ken https://duckduckgo.com/?t=ffab&q=isa+certified+arborist&ia=web

I'll try to take better pictures tomorrow. I'm in zone 8b Southeast Coastal Georgia. We have had lots of rain recently. I thought it might be some kind of fruit because it was growing in the area that the previous owners had made into a garden. We also have a red olive tree and a fig bush in that same area. I'll take some more pictures with close-up of the leaves.

Shiro and Satsuma are entirely different varieties and easy to identify. The Shiro is a small yellow fruit that typically sets prolifically. The Satsuma is a greenish red skinned plum with very deep red flesh. Shiro is self fruitful and does not require a pollinator, but could be a good pollinator for the Satsuma. In answer to your original question i would recommend planting another self fruitful variety that would pollinate the Satsuma if in fact that is the one that remains. That way if it is the Shiro that remains at least the new planting will not need any special variety to get it to set fruit. I would recommend either the Santa Rosa plum or the Beauty Plum.

find some leaves on the ground.. and take a pic of those with a scale.. if you can ... reminds me a bit of redbud or poplar with those heart shaped leaves ... but they are too small in these pix to tell for sure ... whether or not something might fall down.. really depends on whether it will fall on anything ... of value or importance .. this tree looks like it pretty far from anything important.. so why would there be any imperative to have it removed.. btw.. i dont recommend chopping it.. unless you just want some backbreaking aerobic exercise .. lol ... and i presume the car isnt parked there all the time .... in theory.. any tree might fall down.. but we dont really go bare earth to avoid such.. its all about the odds ... and starting with an ID was spot on .... anyway... in tree time.. counted in decades.. ya.. it might fall down.. i dont see any reason why it need be done inside a few weeks or months ... ken ps: perhaps your realtor ought to stick to selling houses.. i dont see any indication from the lawn.. that the tree is aggressive with stealing water .. if it were.. grass wouldnt grow there.. and i note.. there are plants growing under it.. right at the trunk... that doesnt happen with water hog trees .... and btw.. if you decide to have it removed.. thats a bargaining point for a low offer.. if the market will bare such ... on reread.. i see you already bought the house.. congrats .. pps: whats a neighborhood gardener.. and why does he or she have a hate on for this tree ... probably his car parked under it... lol ... or he wants you to hire him to remove it????