Macbook Pro A1278 keeps shutting down/rebooting

First suspect in that particular model is a worn hard drive cable. It suffers a circuitous routing over sharp metal edges in the chassis. With 3-5 years of normal vibration of use and transport, the cable sheathing can wear, causing shorting. Don't bother inspecting it--not all faults are visible--just get a new one.

The cable is available, cheap, and easily user-installable. I have the same model and, when upgrading the mech drive to SSD, I replaced the cable out of caution even though it did not show symptoms. That cable is here:

https://eshop.macsales.com/item/Apple/8211480/

and I used this instructional video:

https://eshop.macsales.com/installvideos/macbookpro_13_unibody_mid12_hard_drive_cable/

The second suspect that comes to my mind is overheating. That model has only one fan so heat can be an issue. Macs are fitted with temperature sensors that signal the computer to shut down if internal temps get too high, roughly 100°/212°F in most notebook models.

Run Apple Hardware Test and post back with any errors it reports. Instructions for your model are here:

How to use Apple Hardware Test on your Mac - Apple Support

While replacing the HD cable, remove the fan and check it and the little heat sink it services for a build-up of lint. I found this in my A1278 after about four years of use:

That little pad of lint was reducing the effectiveness of the cooling system.

⚠️ NOTE: If you remove the fan, simply remove the three screws and LEAVE THE CABLE CONNECTED TO THE BOARD and lay the fan to the right. The connector where the cable engages the board is small and disgustingly delicate. Messing it up means you've ruined the logic board.

You should also check the health of the hard drive using DriveDX. If you connect it to your iMac, then you need to use a special driver for DriveDX to work over USB. If the drive is healthy, then try booting it externally on your laptop to test if the issue is with the internal hard drive cable.

You are most welcome. I'm glad it is something relatively simple.

It's still a good idea to run DriveDX on the drive. DriveDX can alert you when the drive does start to go bad. DriveDX does a fair job of assessing hard drive health, but can get it wrong when it comes to SSDs.

Does this happen when using the power adapter? Make sure to use the power adapter for all tests. Don't connect any external devices unless booting the drive externally.

Try disconnecting the battery and see if the issue still occurs.

What happens if you Option Boot and stay at the Boot Picker menu? Does it shutdown or does the fan run at high speed?

How about leaving the drive disconnected (both internally & externally) and boot into Recovery Mode which will automatically go into Internet Recovery without a bootable drive connected.

Create a bootable USB drive for Memtest86 using Etcher. Option Boot Memtest86 (it will appear as a orange icon labeled "EFI"). In the "Tests" menu I believe you can change the number of loops for the test (located below the list of tests) and adjust it to a higher number so you can let it run overnight.

Attempted option boot, but continual reboot/startup chime continues.

Can you clarify this a bit more? When the boot picker menu appeared, did you leave it sit there without booting further?

Most likely a GPU issue with that screen or possibly a memory issue. You could try removing one stick of RAM to see if it makes a difference. If you still have the issue, then remove the second memory module. Reinstall the first memory module in the original slot from which you removed it. If it still has the problem then the GPU/Logic Board is at fault. Since it occurs so quickly after power on, it is doubtful the heatsink compound will make a difference.

Does the internal Display cable appear to be fully seated?

You may as well try replacing the heatsink compound just in case.

It doesn't hurt to disconnect all unnecessary items from the Logic Board just in case one of them is causing a problem. Even the keyboard can be disconnected if you use the power on pads on the Logic Board.

If you remove both sticks of RAM do you get the missing memory beeps?

It's possible the heatsink compound is dried out and needs replaced.