The very short leaf stalks could point to Quercus robur also. It won't be long before these plants start producing more characteristic leaves and you will be able to tell for sure. Another problem with piggy backing on an existing thread is the party that started the thread may get sent an e mail notification every time somebody posts a comment to the thread.

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????

Cunninghamia lanceolata would be my suggestion, too, based on proclivity to resprout from the base when winter dieback occurs around here. In northwest SC, I doubt winter kill is the cause, but rather the owners might have cut these back on purpose to create this effect.