If I understand this you updated from W10 on a W7Pro64 machine (was that your stock Z620 HP build, or from a MS system builder DVD?) and your USB 3 ports worked in that W10 install. Thereafter you started fresh with a W10 clean install and that did not recognize your TI USB chipset on the motherboard and thus did not place a driver for it in Device Manager as a result.
What I'd do then is a clean install of W7Pro64 disconnected from the internet (to avoid any possible autoupdate), then turn off Windows Update in W7 from the control panel, then I'd use my pre-created bootable W10 64 thumb drive to start the W10 install atop the fresh W7Pro64, and then give it internet access when it wants it.
Plus, I'd contact HP and the W10 team at MS to let them know they have a bug to work out.....
Not sure if you know.... there is a W7Pro64 install bug that results in a very early "update" to the Windows Installer system components, almost instant if you are connected to the internet, that very much messes up the ability of the install to thereafter update itself. There are posts on how to get around this for clean W7Pro64 installs, and my advice is based on a modification of that approach, getting you into the W10 install pathway ASAP without letting the W7install start that process itself. Let the W10 install guide the install/update process.
After your workstation has been registered once with the central MS W10 servers you don't need to worry about activation again. Those servers will recognize your box and activate the W10 install as soon you hook back to the internet after it is done.