This is just a guess, but I suspect that you actually are getting into the BIOS, but that it's coming out of a different physical port on the back of the machine. BIOS on the wrong video port can look very much like a hung machine. Try the usual Del to enter BIOS stuff, and when it appears to have "hung", try connecting your monitor to all of the different video ports in turn (and give it a reasonable time on each, to allow the video stuff and monitor to wake up). Also, since it's Intel platform, try the motherboard video out that comes from the CPU's intergrated GPU (unless your CPU is a SKU with that feature disabled). BIOS video port selection can be odd sometimes, and doesn't follow what Windows thinks is the "primary" monitor. Also, disconnect any secondary displays (and disconnect the cables at the PC end, some have active components in them), try this with just one display around all of the different ports (and do both DP & HDMI, assuming you have both).
Total guess, but it's the one thing I can think of that's a simple-ish problem and doesn't involve pulling the machine apart.
