Mac Pro a1186 shutsdown after booting halfway

@Grant's assessment of a failing drive is still the most likely suspect. You can try booting into Safe Mode to see if it may allow you to boot to the Desktop as it may bypass the damaged portion of the drive.

You can also try running the Apple Hardware Test (AHT) by booting Disc #2 from the original set of disks which shipped with your Mac from the factory (hold the "C" key down while rebooting/powering on the Mac with the disc in the optical drive). Unfortunately the diagnostics don't detect most drive failures, but if the diagnostic reports a problem (assuming all original Apple OEM hardware installed), then the failure is legitimate. If you don't have the original discs, then you can try creating a bootable AHT USB stick to check the hardware using the information in this article (the download links are all direct from Apple's own servers):

https://github.com/upekkha/AppleHardwareTest

It helps to try booting from external media to confirm the basic hardware is functional. Do you have an OS X 10.6 Snow Leopard upgrade DVD to test?

The initial "chime" sound is generated in software when your Mac passes the Power-On Self Test. If the chime occurs and/or startup continues, your Mac is working. The blank gray screen should light up. Then on to the disk Drive.

Accessing the Boot drive:

The solid Apple is not in the Mac's ROM at Cold start. The Apple logo can only appear when it is fetched in the first "blob" of software loaded from a 'magic' place on the boot drive, or re-run after a Restart. Then a whole lot of stuff is initialized, and the progress Bar moves part way across. After a cold start, seeing the solid Apple appear says your drive was able to produce the software that contains it.

If a prohibitory sign appears at this point, it indicates some fundamental part of MacOS is damaged or wrong version.

Mounting the Boot drive:

The next step requires a lot of files by name, so the File System is initialized, and the Boot Drive is Mounted. If the drive directory is damaged, the drive can not be Mounted, so your Mac begins one pass of Disk Utility Repair. This will take an additional about five minutes. During this process, the progress bar may be extended, and will grow by an additional amount not seen on a routine startup.

at the end of that process (which should not take more than about five minutes), it will attempt to Mount the drive again:

-- if the drive Mounts, boot-up continues.

-- if the drive cannot be Mounted, your Mac can do nothing more, so it powers off.

-- if the process stalls, this may indicate you have Bad Blocks on your Rotating Magnetic Boot drive (if so equipped). The re-reading of Bad blocks can take a very long time (on the order of a quarter minute for each Bad Block).

A1198 looks up to retina 2012 to 2015 integrated or discrete graphics.

That happens when your drive is damaged, and even after one pass of disk repair it does not come clean enough to Mount.

FYI, The OP mentioned A1186 which shows to be a Mac Pro (original 2006 or 2008) model.

https://everymac.com/ultimate-mac-lookup/?search_keywords=a1186

Thanks for fixing my screw-up. I stand corrected.