that is the dreaded Paradise, Tree of Heaven, Alley tree.
Let it decompose in the sun a bit before you bury it, or dry for 2 seasons before you burn it indoors.
Roots will travel yards at a time, and if you cut down main tree, the roots will shoot up suckers. DONT let it into your garden area.
https://permies.com/t/9273/woodland/kill-siberian-elm-tree-heaven#97908
You are doing exactly correct with the soil, but would add a few things if you have the time.
Char some of the
wood, then smother to let it smolder without air. Dont burn it in the hole, it will kill any microbes that have survived in the soil till now.
Douse it with some seawater, and if you can, mix in some seaweed with the brush. If you can find a little rock phospate fertilizers, mix some in the dirt you are backfilling.
It appears sea salt has just about a perfect balance of stuff for plants, but salt
water has too much calcium carbonate. Dont use much of either.
Dont leave the dirt you dig out sit in the sunlight, or dry out. Again, will kill any microbes and microlife in the soil that has survived till now.
Order some myco soil innoculant mix from
Fungi Perfecti, or borrow some
compost from an old neighbors pile to get the
local blend started for your area. Drop in a couple scoops when you bury brush.
Try and leave a cross drainage swale (hillock) to capture winter rain, so it will soak down, instead of run off.
A laser level will help with this later. Even a cheapo 2 footer will help a lot.