Okay I'm going to assume a few things here. 1) With gods you mean Apostles. 2) This connection to them is in some way practical and the Apostles and the Spawn are aware of each other and actively do shit together. 3) This connection to them rises with corruption, since you can't even understand monster speech at low levels. You're gonna have a pretty hard time explaining Ella, Nyx and Malik.
Malik is a nat 5 and seemingly as human as they get in his attitude and everything. Probably the sole reason why he's still with us. Nyx is corrupt as fuck and had to use the serum. Ella is both pretty corrupt and abnormally close to the Apostles in every way imaginable. Still couldn't get to 5. Hell, Bernhard had to be told beings of the imperfect Eye's caliber exist by Xanthe. Dude seems genuinely unaware of Apostles or anything higher.
Perhaps it's just that some people adapted to monsterization better than others and that's all there is to it, no mystical connections needed?
I wanted to tackle this in a more broader sense but it was going to be huge, so I'll just answer your points.
The word "connection" has been used under many circumstances. It happens during evolution (Valravn), it allows transfer of power from the planes (Eye during dead end), it allows communication between entities (Originator to spawn, Dreamer to dreamer) and it allows using mental abilities at range (routed through the inner planes). Additionally the level of this connection is kind of measurable, can be somewhat affected by corruption and spawn possessing multiple traits have more of it.
By gods I mean every divine entity, including Apostles, Arbiters, Great Ocean, Henri's forgotten god, the Tentacle thing in the monster world. So far we've seen connections to Arbiters (MC) and Apostles (MC, Michael, Alice). We may have also seen a connection to the Great Ocean (Amber).
Connections don't require corruption to happen. They exist since the spawn's infection. High corruption is described as making you more like a monster and it increases the frequency of the "voices". However there is absolutely no indication that it increases the potency of the effects that use these connections, like mental abilities and evolutions.
Ella, Malik and Nyx are all different.
Ella:
Meeting her after killing Jake, she says that level 5 is when the monsters know you are ready. Ella was deemed a failure by Memory when she completed her 3rd, so she will never be ready.
Nyx and the rest serum level 5s:
HERO agents avoid natural fifth evolutions because it's deemed dangerous. This is explained by Clark. So not only you have to actively look for some arbitrary requirements but they can kill you but completing them does not necessitate you will evolve, it just unlocks the potential.
Clark "There's a bunch of requirements leading up to it that'll probably kill you before your evolution even starts. Henri met his on accident, or so he tells me. Bernhardt and Malik sought them out afterwards."
Clark "But yeah, to answer your question, they didn't want us trying it. The evolving on its own is dangerous enough as it is, and even fulfilling those requirements don't guarantee it will happen or that you'll survive."
Malik (and the other 2):
Considering Ella's words that I mentioned earlier, the thing that makes the most sense to me is that these 3 have different (and easier) requirements. Consider Henri for a bit. If you are a forgotten god with a mangled line you can't be too picky about allowing your "Chosen" to claim your power. If on the other hand you are an Apostle, you have too many options. Even if your 100% purity Chosen fails, you can probably find a 90% purity one strong enough to do what you need to do.
And finally, the reason why I don't think they are just people who adapted better to "monsterization", is because the Eye recognises them and knows who they are. It's obvious that the Eye doesn't know who Malik is. He knows who the "Completed, but kept whole by faith" is and that he shouldn't be allowed to thrive or coexist with the other 2 for some reason.
When the Eye speaks to MC, he says this:
Unknown "I trust you'll figure it out soon enough."
Unknown "I mean, who can you trust, if not yourself?"
This points to the Eye considering itself and MC as the same entity and extends the same logic to the big trio.
