I didn't give up on my bathroom floor. I've tried things little by little and seem to finally have hit upon a winning combination: SOS pad (used in moderation) followed by a Clorox wipe and then a buffing with a cotton cloth.
I'm not done yet but making progress. I'll let you decide if it's working. There are about five different spots where I have deemed the haze "really bad." Here's one just inside the door.
It's one of the parts that I have focused on because it is just inside the door -- marring the beauty of the rest of the floor at first glance. Here's what I've managed to do thus far.