I've plotted several locations based in Auckland, New Zealand, using Grid NZGD2000:ESPG:2193. When I load QuickMapServices and set to the same grid, The basemap produced is situated off Omen. 
[GIS] QuickMapServices not showing basemap for the location – QGIS
basemapqgisquickmapservices
Related Solutions
Your points disappeared because you not specify Spatial Reference at last step adding csv to the map.
Your csv data are in WGS84 and base map (i.e. Google or Bing is in Web Mercator). Also you need to enable reproject on the fly.
I was utilising the correct projection (obtained from http://gpsinformation.net/utm-zones.gif) but I needed the project CRS to be transformed to this projection for my mapping units to move to meters.
To transform the project CRS, as opposed to 'on the fly' reprojections, I saved a new project with my desired projection (WGS 84 /UTM zone 59S EPSG:32759).
Then followed the same workflow as before: Vector -> Geometry Tools -> Export add Geometry Columns.
The key point of difference:
As suggested by Kazuhito reprojecting each layer by resaving them in the desired projection would enable the use of Layer CRS.
Best Answer
Per the comments, you're doing it wrong.
You can't simply change a layer's CRS if it was defined as another.
How to resolve: Make sure your layer in the original EPSG:2193 CRS.
If after you load the basemap, the canvas doesn't change to EPSG:3857, change the canvas to EPSG:3857, and zoom to your layer's location in New Zealand.