It seems that most applications I use for serving map data, gdal2tiles or Geoserver takes the supplied GeoTiffs and chops it up into png files with no metadata. What is the standard practice to place these tiles correctly. Do I just lose elevation and bounding box data and that's it? Do I just place the tiles based off of the transform of the previous zoom level? I feel like there is something fundamental I'm missing.
It is not quite true that gdal2tiles does not create metadata. If you use the default profile then tiling schema is the one used by Google Maps and it is supposed to be well-know. If you run gdal2tiles with raster profile (-p raster) you will find file "tilemapresource.xml" which contains the necessary Tile Map Service metadata. Individual tiles do not have metadata and their georeferensing is interpreted by the metadata of the tiling schema. Tile map clients know how to reach each tile and where to place it on the map.
GeoServer is sending tiles through a service and users are supposed to know how to access the metadata from each service. Those requests are like
Live example about WMTS metadata
Save response on disk, it won't open into your browser.
I don't understand how elevation is related with tiling and metadata.
You need.to use gdaldem assign color values to elevation ranges then run tiles. You can also use JOERD python script by mapzen that create terrarium PNG format.take a look at geopackage elevation that probably your best bet