Do the Greens really need to care what National supporters think?
The only kind of deal the Greens could accept would be something that addresses everything National has failed to do on the environment or on social issues in the past 9 years. Not vague promises of studies for future implementation, but real action now. Ain't gonna happen.
I think the Greens need to care what the media report, and since the standard "analysis" is usually to get some Nat-aligned commentator to give reckons (because free and easy) then, yes.
If that didn't matter then Metiria Turei would still be co-leader. The Greens can certainly say "don't care", and I'd be fine with that, but they shouldn't say "don't care, but will later after entirely predictable pressure".
I think the Greens need to care what the media report
Yes and no. I'm not a Green Party insider, but especially after Kennedy Graham and David Clendon resigned, it seems to me like what happened to Metiria Turei went deeper than just media. For a moment that came from right within the core support base of the Green Party, which apparently felt as if it hadn't been adequately consulted.
Anyway if the Greens did risk extermination by going into a coalition with National, I can't imagine it'd be a stable government long term. More likely it'd go the way of the Land and Water Forum, where a bunch of high profile environmentally-focused NGOs ended up leaving because they thought their input was being ignored and their presence was merely being waved around to make the government look good.