Let's hope it isn't the GPU, so I'm going to suggest another diagnostic tests. I have two ideas:
a) Try booting into single-user mode. That mode doesn't use a GUI or load video drivers, so if you can see the apple logo on startup but single-user mode also fails then the fault is probably something else.
https://support.apple.com/en-us/HT201255
https://eshop.macsales.com/blog/42126-safe-mode-single-user-mode-what-they-are-how-to-use-them/
b) Try booting into MacOS or recovery mode on an external drive. If this works then there may be file corruption crashing your main startup. If a second boot method also fails then yah, it's probably hardware.
...
Anyway, there are no good GPU replacement options.
2010 iMacs use standard MXM3 laptop-style GPUs (I think the A1312 - 2390 has a 2-heat-pipe MXM-B) but Apple never made any official upgrades/replacements. That limits you to either a used 2010/2011 OEM iMac GPU or else gambling on a PC laptop card.
Used OEM cards have no software/hardware issues but are old and expensive. It would almost be cheaper to buy an entire used 2010/2011 iMac than a working GPU. Still if you only want a GPU I'd suggest another 5750. The 5670 and 6770M are slower while the 6970M had reliability issues.
PC Laptop cards are cheaper but quirky. Most are unusable until flashed with a custom a iMac VBIOS (amateur developed, no official support), but even then some makes/models just fail. GPU temperatures also aren't properly recognized by MacOS so 3rd party fan control software is also needed. A Quadro K3000M or an MXM AMD WX4150 is probably the fastest that could be installed with the original heat sink, but you would need to do a lot of research on installation/flashing and find a custom vbios.
