I understand that the grip of the Z6 III holds the battery. Judging by pictures, it also holds the memory cards - which includes a UHS-2 full-size SD card instead of a UHS-1 micro-SD card like the ZR has. Oh, and you can also change the cards while the camera is on a tripod or gimbal.
I had no idea that mirrorless cameras couldn't be used without a gimbal. That's quite a limitation. Without being aware of it, I've been using mine mostly without a gimbal for a while now.
You know the microphone is supposed to point toward the front of the camera and not the back, right? That should reduce the risk of poking out your eye with it when holding up the EVF. 😉
Anyway, the Z6 III has a screen that is still pretty decent-sized at 3.2".
From your description, it sounds like your needs could be better met by using a Black Magic Cinema Camera 6K instead of a Nikon ZR. The BMCC has a bigger rear screen at 5". The EVF is optional so you won't need to use it if you don't want to. While L mount isn't quite as versatile as Z mount, it's still quite good. The BMCC6K also has 2 mini-XLR mic inputs which can be used along with the internal mics for 4-channel audio. If you want smaller files, you can increase the compression ratio for BM raw. Even at the maximum compression on the UC12K (12:1, I think?), it still looks very good. I usually use the Ursa at 8:1 except for VFX shots where I go up to 5:1.
The main limitation against the ZR is that the BMCC is a bit bigger and heavier. But as a person who wants "the best back LCD I can have," the BMCC blows the ZR and its puny 4" screen out of the water. You'll also get a full-size HDMI port and not a garbage micro-HDMI port in case you want to connect to an even bigger 7" screen.