I've found a solution. To generate new GeoWebCache tiles (out of a new raster file added to an ImageMosaic Store) you need to:
Remove all the index files (shapefile and the rest) that GeoServes creates in the main folder of the ImageMosaic
Add the new raster file to the ImageMosaic folder
Go to Stores and hit Save to create a new shapefile and the other ...
GeoWebCache has no idea that you've updated the index file or the mosaic, so you first need to delete the tiles that fall within the bounding box of the new imagery and then ask it to generate missing tiles.
If you want to use browser to view files on file system on your local machine, then for security reasons you cannot refer to those files by absolute path, you have to use relative path.
In your case that would mean that you have to put for example myTileViewer.html file with your code somewhere on the path to tiles, let's say in the folder C:/Program Files(...