Problem
With Windows updates starting from August 10, 2021, Point & Print restrictions were introduced that no longer allow normal users to map printers whose drivers are not yet installed on the client side, because the required drivers may no longer be installed automatically in the background in the user context. This also applies to drivers already installed on the client side, which would be automatically updated during the mapping process if newer versions were available on the print server.
Solution
If you want to leave the Point & Print restrictions as they are for security reasons, you should make sure that all drivers of the shared printers are already installed on the client side (e.g. terminal servers) in the same version as on the print servers, otherwise the mapping of these printers as users will fail.
Alternatively, you can either uninstall the updates or explicitly disable the new Point & Print restrictions that allow printer mappings (or more precisely the driver installations or updates in the background when necessary) only with administrative privileges. You can disable this new default behavour by setting a registry key (see link below). In this case, you could at least gain some security benefits by explicitly specifying the list of print servers from which users are allowed to map their printers from. More information about this can be found here: KB5005652—Manage new Point and Print default driver installation behavior (CVE-2021-34481) (microsoft.com).
An overview of Point and Print in general can be found e.g. here: Introduction to Point and Print - Windows drivers | Microsoft Docs
General Background Information
We have known since 2019 that some Windows Updates cause printing issues (e.g. spooler crashes and breaking the printing capabilities of Windows). For example, KB 4522016 from 10/2019 or KB 45411329 from 03/2020. Microsoft usually fixes these issues quickly with a new KB Update afterwards. As a workaround you would have to uninstall the problematic KBs temporarily.
Update March 2021: Windows Update KBs from 03/2021 are causing a wide range of printing issues (e.g. blank or false output, spooler crashes, BSOD and breaking ThinPrint v-layer configurations. The issues are reported to sometimes be dependent on the printer driver. See e.g.: Windows 10 KB5000802, KB5001567 causing new critical printing issue (windowslatest.com)
Update July 2021: Microsoft recommends restricting certain Windows Print Spooler functionalities to counter a vulnerability known as "PrintNightmare" (CVE-2021-34527 - Security Update Guide - Microsoft - Windows Print Spooler Remote Code Execution Vulnerability)
Update August 2021: Microsoft recommends installing a security update that changes the default Point & Print behaviour (CVE-2021-34481 - Security Update Guide - Microsoft - Windows Print Spooler Remote Code Execution Vulnerability)