Supported! Although, I think Hatchling Rush is okay as is, in 2v2 it is actually the best primary core, even better than lifeline.
I don't know exactly what the ED team envisioned for cores when they invented them, but to me they need to play a more central role in the build making process like the way actual skills are. I like how that's the case with robot cores, where you wouldn't ever choose a random robot to use for your build. And a lot of times people make builds that even revolve around a particular robot, or at least depend on it for the build to be any successful. Right now the only example of a build that revolves around an equipment core is Spirit Meld. With equipment cores, for the most part it just seems like people choose among one or two of what's clearly the best regardless of what build they intend on making since there's such a disparity in their power, and many of them like the Boosters/Generator basically just act as one-time hacks instead of being a meaningful part of a specific build.
I support this idea as well, IF cores start costing less to buy or you keep them when you put a new one in. Otherwise, if cores become central to builds, the cost of changing builds goes from a simple 250 + 100 per item stats changed to that + 10,000 per core needed to change.