If it were my tree... I would dig it up and get rid of all the old compacted soil around the root ball. I would make sure the roots are nice and loose. And dig a big hole and mix up some soil and amend, mulch heavily. the point is the roots need attention. Make sure the soil drains well...

Not enough to go on - esp. the pieces that have been dead for some time

Post some pictures. Closeups of leaves and bark, also a view of the entire tree.

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????