The link between saturated fat and LDL cholesterol is extremely clear. Far from coming from low-quality studies, the relationship has been observed in RCT metabolic ward studies. The best data you can get on the acute effects of diet and we are talking hundreds of these experiments all observing the same relationship.
Does high LDL cause heart disease? That is probably less clear, I am not a scientist, so I really don't know what the latest is on this. LDL is present in Atherosclerotic plaques so it's not crazy to think it plays a role. If I was to guess, as a complete non-expert, low-fiber intake and inflammation probably play a role in turn LDL into atherosclerosis.