I don't want to hijack someone else's thread, but I just came cross this same issue. Worked fine in 3.4, moved to 3.6 and no Basemaps button. I'm using my own basemap services and have several defined in the Basemaps tag. I don't see any issues in the configs, have read and dont' see any new changes in 3.6 that should be relevant.
The layer tags themselves seem to be fine - if i set any one of them to visible="true" they display okay. Just the button is missing.
Greatful for any ideas as this is my last piece to being able to complete the move to 3.6.