After many hours of troubleshooting, I've found that disabling the MU-MIMO function on the R7000P gives my devices a stable connection.
Advanced > Advanced Setup > Wireless Setting > [Disable MU-MIMO]. Apply.
I have been able to transfer several large files from a wifi device to a wired device without constant dropouts/disconnections at the expense of actually being able to use the MU-MIMO functions of the router.
There are either issues in the way Netgear has implemented MU-MIMO or an incompatibility issue with some other wifi devices on the network. Since I replaced the r7000P with a new one I suspect it's an issue in the Netgear router not my devices (I can't confirm this though as I don't have a different make/model MU-MIMO router to test).
Since MIMO is one of the main selling point for this router this also raises a few new questions.