Excellent investigating, ChronicQuery! I think you're really on to something.
With what you've put forward, I think the person who helps Argos on the ruthless route directly has our answer about the Bedrock and 'Obsidian': (Some spoilers expunged since it's a little out of scope for our topic at hand 😉)
On the night of [this realm's] creation [the gods] offered [their] ichor as ink for its constitution. First a droplet from each of the twelve atop Olympus, who cast their votes in the hybrid's trial. Then five more from those denied a say in the matter [...] Each drop carrying the echoing will of its god, mixed together to write the realm's legal bedrock.
While it's easy to assume the ink for the constitution is literal ink for the Primordal Contract we find, its description outright says otherwise:
In truth, these pieces of parchment are merely a copy. The original Constitution of the Labyrinth does not exist in paper and, in fact, was never meant to be read by mortals. Its original format is beyond human understanding, and even today lies hidden from sight.
I think the obsidian at the bedrock (our ruthless route friend deliberately says bedrock above) is the mixed together blood of the gods, and its interior contains the real constitution which also powers the realm. It is, after all, the literal bedrock the realm is built on.
To further back this up, also in the ruthless route, a certain location will not work because it does not have salt, as salt is one of the forbidden substances to be generated by the realm. The friend gives Argos "a powerful artifact, the only one of its kind within this realm", and at the location, the friend says "That artifact I gave you shall be a sufficient substitute." Here's what happens when Argos uses the artifact:
He unwraps the artifact [...] from the cloth that swaddles it, freeing that hunk of cursed rock from its hiding place. Dark as obsidian. Dark as the still waters stretching out within that underground cavern. Dark as the ichor which runs in the gods' veins. It was no easy matter to steal it away from its origin deep within the bedrock of the realm.
I suspect it's a specific god's drop of blood, pulled from the obsidian in the bedrock, and it even gets compared to divine ichor because I think it is divine ichor.
If all of this is is true, it explains why Asterion doesn't want us poking around: he still wasn't sure if he could fully trust us back then, and it's just not safe to mess with if it's the literal magic stuff that's keeping this realm together. Heck, if it's supposed to be beyond human understanding, it might hurt our brains to try and observe it, so Asterion is also keeping us safe by just covering it up as 'unclean obsidian' that we should just leave alone.
It might also explain why the MC can hear the bedrock grinding more than Asterion when we go down to visit it the first time, this artifact , no matter what it is, might be trying to make itself known from in the crystals, now that we know it's hidden in there from the ruthless route. It would be a great place for a different Chekov's gun to come into play if this artifact is dangerous to access or remove, but that's not really tied to this topic of blood so we'll let it lie for now.