Wow, that's very unfortunate that all of the add-in cards are glued into their slots. I have built and serviced many computers over the last 10+ and I have never seen that before. Maybe I was not going into the case to replace something that was glued in and just did not notice. That's too weird... even stupid.
That sounds like something Apple (or the old Packard-Bell company) would do to force you into buying a new machine instead of being able to upgrade anything. What brand of PC do you have... I am dying to know who did that.
One last suggestion before recommending you send your PC to the recyclers dump; and, since you say the problem is present during POST it is probably going to be a waste of a few minutes. Have you tried a bootable CD? If you get the same result with your video from a bootable CD, you can conclude 100% that your system (due to the glued in video card) is ready for the scrap heap.
It sounds to me like you are definitely running out of options. Since you really have nothing to lose with the prospect of replacing your system, or buying a new mobo and all new internal components, I would considered trying to remove, or cut away the glue, to free the video card and other components. If for no better reason than to defy the wishes of the evil manufacturer, it would be a worthy effort.
But, if you are going to buy a new PC instead of building your own, don't waste your money on a video card.