You definitely want to replace the “black dot”, you can usually get them from another, dead, home button. That is why the click isn’t working when everything is mounted.
As for the bracket, I find that using just double-sided tape makes for a squishy feel in the best case and outright slippery in the worst. I would start by properly cleaning all of the old adhesive aways, on both the screen and the bracket. Then apply good quality tape (3M/Tesa). Once the alignment is good and the button works properly, I apply a small bead of epoxy around the bracket (not on or near the flex!) - this is what Apple has been doing of late. This locks in the bracket and should make for a nice repair.