Provided single site admin permissions is not enough to view all settings and try to reproduce the issue. Superadmin access is required.
Btw., I do not see it at your older video, – what role was granted to the user armel at the ‘lite’ subsite when you recorded that video?
I got a thought about possible confusion of pmpro level ‘lite’ and user role ‘lite’, for which you added admin menu access restrictions. Definitely pmrpo level is not a user role and vise versa.
2nd, after more careful look at the same video, I see that you did not really fulfill “Network Update”. When you clicked “Update” from admin menu ajax data exchange with server took some time and your next click on the “Network update” button did not take effect as URE Pro just waited/received ajax response from a server and showed notice at the top about “admin menu access data update” for the current role. It was made for the main site only. When you will click “Network update” you see 1st the dialog where you can select what add-ons data to replicate for all subsites. Look at this screenshot.