For those playing at home: Altium, apparently, introduced "component" pushing in 16.1. And this was enabled by default.
Preferences -> PCB Editor -> Interactive Routing -> Component Pushing -> Ignore
This was probably the only parameter on that page that I hadn't tried.
Also: kudos to Transfer BV. Took a 3 min phonecall. So, I guess my subscription almost payed off for this year O0