So I'm playing whack-a-mole with notifications stating
process com.google.process.gapps has stopped and
Unfortunately, Google Play Services Has Stopped. All seem to be stopping me get past the connection check screen.
I've booted into 'Recovery mode' and wiped the cache partition, but to no avail.
I've one more suggestion from the Nexus help forum (thanks Penelope), but it looks like my 2013 Nexus7 is going to be laid to rest.