Well, it's certainly something you could try, especially given that the alternative is a rather expensive full screen replacement. As Alec already mentioned, a professional quality repair for the front glass requires a couple of pieces of equipment that you and I don't have lying around in our garage - a vacuum chamber and a UV chamber.
To start with, even professionals can't claim a 100% success rate just removing the old glass; it's difficult and all too easy to break the screen in the process. After that, you apply the Optically Clear Adhesive, or OCA, glue to the screen and put the front glass on. Now in order to get all the bubbles out, you stick it in a vacuum chamber, then it's ready for the UV chamber to harden the glue and make the connection permanent. Here's a video showing a guy who seems to basically know what he's doing making that repair to your phone.
Samsung S23 Ultra - Broken Screen Replacement - YouTube
To be honest, I've attempted this myself with my brother on a couple older model phones and the results were mixed at best. One of them looked mostly okay while the other wasn't all that great. Usable, but you could definitely see defects in the glue layer; kind of like a screen protector with dust particles and air bubbles in it.
So yeah, it may well be worth a try; that's your call, but I suggest going into it with modest expectations. You might want to go ahead and preselect a full screen replacement just in case the results aren't something you can live with.
Normally the professional quality screen replacement on your phone is done by replacing the complete screen + frame assembly, so you're basically transferring the guts of your old phone into a new chassis. That means the screen is preattached to the frame, alleviating you from any worries about sealing and securing the display to the frame. Here's a guide from the Spanish site Nadie Me Llama Gallina showing you how to do that replacement.
Manuals / Samsung Galaxy S23 Ultra / Full screen
Good luck with your repair; I'd be interested in seeing the results if you do choose to go ahead with the glass replacement. And if you're feeling a little bit ambitious and altruistic, you can take lots of pictures as you do your repair and use those to create your own iFixit guide to screen repair/replacement! It's easy and fun and the community will appreciate you!
Create a new Guide - iFixit
By the way, with regard to the rear cover, just buy a replacement and don't even try to mess around with replacing the broken glass. They're cheap enough that it's not worth your time and I don't even know if replacement glass is even available. I'd suggest trying Amazon, eBay and AliExpress depending on your budget vs. time frame considerations. Here's one that came up first on Amazon.
Amazon.com: Galaxy S23 Ultra Back Glass Cover Housing Door Replacement for Samsung Galaxy S23 Ultra 5G Back Glass Cover with Pre-Install Tapes (Not Suitable for S23 / S23+ Plus) (Phantom Black) : Cell Phones & Accessories