The wifi network requires a WPA2 password

The original A1186 models (all 2006 thru 2008 models) of Mac Pro silver tower had a Wi-Fi card Option. It was not necessarily installed on every Mac sold. to check if yours has the card, use:

 menu > About this Mac > (System report) > ... Wi-Fi is in the second group under Network heading

You should see (after a palpable delay) a display that shows if you have a Wi-Fi card, and what channels it supports.

You can buy a replacement card from the usual suspects. Installation is tricky because the parts are very small and the antenna wires (already present in your mac) are fiddly.

Installing the wrong card may preclude certain system updates in the future (possibly because your Mac does not seem like a "real" Mac.).

Keychain Overview

https://support.apple.com/kb/PH20093

Won’t harm anything, it will restore a new up to date copy once you connect to the network again.

As you can see the network was joined the first time in Keychain back in 2016

So they still appear in the preferred network window?

Did you “apply” after making the changes and reboot?

Did you manage to

Open Keychain Access

located in > Applications > Utilities

Then select Passwords

on the left.

Type in the name of your Wi-Fi network in the search field top right corner of that window.

Then right or control click that keychains you listed previously and Delete them all.

Reboot the machine and test again

You have hardware, good!

Depending on the software versions, all references to "Airport" have been replaced by "Wi-Fi" going forward, especially when the card (as yours does) has dual-band (2.4GHz AND 5GHz) capability.

System Preferences > Network > Wi-Fi > [click the lock; enter your Admin name and password]

The lock on the Wi-Fi pane only needs to be unlocked to make changes or see (advanced) items.

To troubleshoot further, hold down the Option key while you click the Wi-Fi icon on the menuBar. You get a display similar to this old one:

what do you get for PHY Mode and channel?

what do you get for signal or RSSI, and at what distance from Router?

what do you get for Transmit Rate?

and very important, how many other networks do you see?

Your next step might be to set it up again from scratch.

system preferences > network >Wi-Fi > (assist me) > (assistant) ...

... be sure to specify a NEW network-Location-file, not "Automatic", and give it a meaningful name. This will discard all pre-conceived notions and use the answers to your questions, the environment in which you are running, and sensible defaults to produce a new setup.

Open network preferences, on the left select WiFi and on the bottom right select advanced.

You will see preferred networks with a list of networks the Mac connected to in the past,

Select all the fios networks and delete them using the minus sign.

Chances are the keychain of which stores the network name on that computer got corrupted and you need to find it in the keychain access app on the Mac. Either this or change the network name/s.

Open Keychain Access

located in HD > Applications > Utilities

Then select Passwords

on the left.

Type in the name of your Wi-Fi network in the search field top right corner of that window.

Then right or control click that keychain then click Delete.

Reboot the machine and test again

system preferences > network >Wi-Fi > (assist me) > (assistant) ...

This step invokes a setup wizard.

It asks a few EASY questions about how your network works, (like do you dial out ?) and completes the rest by looking at your hardware. it is not at all onerous.

That USB adapter will turn out to be MUCH more complex to Install and set up than the simple auto-setup assistant.

You can also get a FREE evaluation and setup help at any Apple-owned store, provided you make a genius bar appointment. They claim they do not have parts to make repairs any more, but you do not need any parts.

Those are the networks you connected to in the house hold. Delete them all and reboot.

Afterwards try to use assist me as mentioned by previous advisors.

That pane shows you on Channel 1 (in the 2.4GHz band) with about the best signal strength possible, -44. You must be right on top of your Router -- maybe even too close (you should try for 3 to 6 feet, if possible).

With this really good signal strength, and with the ability to determine that you are using 802.11n, I do not think your antennas or anything else is broken.

But the Transmit rate of 1 and MCS Index of 0 are both wonky. Oh, it says no Internet connection. That may be a setup problem.

I would NOT be in a hurry to replace the card you have now with one of those crappy little USB things.

Is this in the physical location where you expect to use this computer? i ask because you appear to have access to the 5GHz band under a different network-name. But 5GHz does not reach quite as far.

if he can’t connect in the recovery it’s his router bottom line. How about creating a Hotspot from his IOS device and see if he connects?