I very much want to use the automated straighten feature, sadly the crop cuts part of the image and it absolutely cannot... Is there a way to use it without cropping? I really can't understand why it wouldn't be an option to disable 1 or the other, surely it can't be sharing the same logic and would be pretty easy?
This feature would probably make quite a few people happy, and it would alleviate a huuuuuuge amount of time I waste straightening images manually or using the tool, but why do that if there's an automated one you guys already created?
If you don't like how Photoshop automatically fills in the missing pixels, another approach is to distort the image to fit the new document boundary with Free Transform. If you have just a small are of transparent pixels then Free Transform > Warp will often work letting you pull a corner to fit the image to the frame.
You didn't mention Perspective Crop. That can be used to automatically crop, straighten and square up an image to the frame. I can't help but think the tools are already there. You just need to find them.