Major bug in ArcGIS Zonal statistics?

There is a bug that seems to correspond to what you're experiencing - it's registered as BUG-000084883 - The 'Ignore NoData in calculations' option in Zonal Statistics as Table tool {and Zonal Statistics tool} is not honored when checked off, producing incorrect results.

It occurs with 10.3 and 10.2.2 but not 10.1. Did you try the tool with this version?

It is a bug. Something terribly wrong with cell count.

Correct mean (9.0452380952381) times correct number of non-empty cells (420) divided by 297 (that is a cell count reported by tool) results in 12.7912457912458. That is a wrong average reported by tool.

Results of my own toy size grids test: