Welcome to the forum, Antoine!
Yeah, the use of CLR was a rather large mistake. It appears to have severely etched the surface. There’s no “cleaning” that will help. You appear to be on right track with wanting to polish the surface to remove the micro-fine rough, etched surface to restore an extremely smooth, polished surface.
Polishing the surface of tile isn’t hard using wet polishing pads and water. But it’s hard to avoid a wavy surface with a small, single pad. Usually, it takes a lot of practice and more advanced equipment. And it’s rather challenging containing all the water.
The floor tile appears cracked. If that’s the case, there’s no halting that damage.
I’m assuming you’re just trying to make it better while you’re there and looking for any helpful info you can get your hands on? And do I dare ask if the owner has given permission to attempt repairs?