Yeah that makes sense. The rationale behind the way I was taught was never go over them, cause you never know their history with being hit, close your hand, so if they are aggressive, they don't bite off your fingers, and angle down, keeping your hand below their snout, so you are non threatening.

As someone who has now owned a very reactive dog and worked with trainers, and been bit while interning at the vets, the best thing to do is actually just ignore the dog. If the dog starts leaning into you, showing loose body movements, and typically large tail wags, these are pretty good signs they might want to be pet. As you also mentioned, don’t look over them; additionally, no fast movements, and you really don’t need to offer your hand - they can smell you from where they are.