Mac Pro (Late 2013) installer resources have expired and booting into safe mode shuts down the mac.

These Macs are New-to-You?

Verifying their eligibility will be done with the Apple-ID recorded on the Mac. That is the Apple-ID of the previous owner. if you don't also have their Apple_ID password, you won't get far.

take a look at this procedure:

What to do before you sell, give away, trade in, or recycle your Mac - Apple Support

First, boot into Recovery and repair the disk. if it can not be repaired, it will need to be erased before proceeding.

You did not state where you got your installer from, but it appears to be out-of-date, and you will likely need to download again.

MacOS is not copy-protected in the usual sense:

Apple software downloaded from the Mac App Store, including MacOS, is not copy protected. However, it IS tagged with the Apple-ID of the purchaser when "purchased" (or equivalent. Think of MacOS as purchased on sale for $0.) A valid Apple-ID is required for almost all downloads from the Mac App Store.

You cannot update, upgrade, or install new software based on already-installed software UNLESS you supply the Apple-ID of the original software.

If your old MacOS is damaged, or you completely erased your Mac, the Apple-ID is not available, and you get the “cannot contact the server…” message that should [but does not] go on to say, “…with the Apple-ID provided or NO Apple-ID”.

In that case, you invoke Recovery with:

Shift-Option-Command-R

You are requesting the "shipped-in-the-box" version, which is tagged ONLY with the serial number of your Mac, and does not require any Apple-ID to be already working on your Mac.

Invoking Recovery with

Shift-Option-Command-R

should provide the Original 'shipped in the box' version of macOS.

if you are being offered Sierra, that is NOT the original shipped in the box version.

on that screen select the very TOP item,

Apple SSD SMO25...

and ERASE.

New format offered should be GUID partition scheme and MacOS extended (Journaled). choose a Name that you like or 'Macintosh HD'.

as I mentioned, 'MacOS Base System' (of size under about 2GB) is the name for the minimal MacOS used to support Internet Recovery, Disk Utility, and Installer. In this case, this Disk Image is pulled from the Mac's ROM and Mounted for your use. It is currently running your Recovery session. MacOS Base System cannot be Erased.

Now that the DEVICE has been completely erased,

Invoking Recovery with

Shift-Option-Command-R

...should provide the Original 'shipped in the box' version of macOS. There should be no need to involve any Apple-ID.

This does not seem to be going where you want it to go. perhaps another approach would be more productive.

If you have another Mac of a similar era that is up and running, there are several different ways you could proceed instead.

create a bootable USB-stick installer.

NB>> this is MUCH more complex than just dumping the files onto the USB stick. The USB stick must be turned into a BOOTABLE installer.

Create a bootable installer for macOS - Apple Support

Also the downloaded MacOS images are in a disk images, which muist be Mounted, then the Package must be run to PLACE the Installer in the /Applications Folder of the Host USB-Stick-making computer, THEN the create install media sequence must be run.

follow faithfully the instructions on this site for 'Use Web browser'

How to download and install macOS - Apple Support

Two Mac solutions:

With certain combinations of new and old Mac, you can use Target Disk mode to repair, erase, and install on the drive of the old Mac, by treating it as a disk drive on the new Mac.

Transfer files between two Mac computers using target disk mode - Apple Support

This article describes transferring files, but the full gamut of operations is available (including installing of any MacOS supported by the HOST computer) and erasing the drive.

2013 models support 10.8.2 thru 10.15 Catalina

2016 models support 10.12 Sierra through MacOS 12 Monterey

.

<< But its currently stuck with an account lock >>

what does that mean exactly?

can you boot into Recovery? that Mac does not have a T2 chip interlock.

Macos base system of size of 2GB or less is the recovery software. It is in the macs ROM.

For this purpose, just erasing Macintosh HD will NOT suffice.

You must choose "show all devices" and select the immutable manufacturer-given Device-name.

then ERASE the entire Physical DEVICE.

Restart your Mac and immediately hold these keys down, and keep holding them until the display stops changing

Shift-Option-Command-R