Install of IIS role and Windows Update both fail

I have a client that uses a database application called DocSTAR. They recently had to upgrade their server OS because the new version of DocSTAR doesn't support older versions of Windows Server. We got them new server hardware so that they could have their SQL server separate from their domain controller server, and installed Windows Server 2016 on it. After getting SQL installed and getting the new server physically installed and put onto their domain, we then contacted the company that distributes and handles DocSTAR, letting them know they could install DocSTAR and transfer the data over to the new server.

This is where the issues came up. While trying to install and configure DocSTAR, DocSTAR's installation also tried to automatically install IIS. This failed, and looking up the error in the CBS.log mentions that it is because the files could not be found (I'm sorry, I forgot to document the exact error code, my fault). Tried manually installing IIS through the Feature & Roles GUI, same error. Research said that it's possible that it was failing because there were updates installed that required a system reboot to be applied before the role install would work. Tried restarting the server, didn't fix the issue. Noticed that there was a update waiting to get downloaded and installed, so figuring the update might fix the issue, we tried installing the update.

Didn't work, error 0x80070003. Researched this error, and tried doing an sfc /scannow, tried stopping the update service, clearing the temp files and downloads, then starting the service and trying again, didn't work. Tried the dism /online /restorehealth and that didn't fix the issue either. Now I'm getting error 0x80070002.

If I remember correctly Windows Update and the Features & Roles installation GUI use the same underlying systems, right? So if update isn't working, installing a new feature won't work, so I've been concentrating my efforts on fixing the Windows Update, but I can't seem to find the solution.

Any help is appreciated.