So, if a nascent company chooses to go down this same path of generating (or maybe _seeding_) their logo design with AI, have they essentially given up any ability to protect that logo going forward?

Logo's are generally protected by trademark rather than copyright. I don't think anything prevents you from using a generated logo with trademark. For example you could have a trademark on an orange square, even though you could never copyright it. In the same way a trademark could protect your product name even if it is a single English common noun, as long as it is distinctive in use within your trademark scope.

This is kind of a weird take to me given that photoshop exists. (Tons of proto-computer vision algorithms in there, like basic convolutional filters.) I suspect you'd still get copyright if you modify it a bit somehow.