Sorry to reanimate this thread but I ran into a similar issue and this thread was one of the first results when searching.
I solved this by looking at my Altium rules. There is a rule under Routing -> Routing Layers and this was set to only allow routing on the component layer. I didn't create this but I did use one of the Altium pre-sets for a four layer board so maybe this gets created automatically?
Just posting in case this helps someone else with the same problem in the future.