If file size is your worry and accuracy not, maybe use a lossy format for tiles like JPG or JP2 rather than PNG?
Worth noting DEMs are moving away from tiled formats recently. Mainly to COG (Cloud Optimised Geotiff) which isn't the most efficient but is a simple tweak to a file format already broadly adopted. There's a few others out there aiming for efficency at scale too - ESRI has CRF and MRF for instance, but nothing has become industry standard other than COG yet.
Worth noting DEMs are moving away from tiled formats recently. Mainly to COG (Cloud Optimised Geotiff) which isn't the most efficient but is a simple tweak to a file format already broadly adopted. There's a few others out there aiming for efficency at scale too - ESRI has CRF and MRF for instance, but nothing has become industry standard other than COG yet.