[2022] I just can't install Paradox Launcher v2

That second screenshot shows the download for the standalone Paradox client, aka launcher v1. That is a separate application for a different purpose, is nothing to do with your problem, so you can ignore that.

launcher-installer-windows.msi is in your Steam-Stellaris folder.

In your quote from my clean install process you have missed the most vital step., which is running steam.exe as the Windows Admin user. Please try that.

IF that doesn't help, in this folder are some .log files:
C:/users//AppData/Local/Paradox Interactive/launcher-v2/
Please attach them here.  

The 1603/1605 errors we see there are typically a previous other installation stuck, or a problem in the Windows update/install processes.

If you see msiexec running in Task manager-processes, kill it and try this launcher install again please.

Otherwise run the Update app in Windows, does it load okay, find anything to apply - whether a patch, update, or whole new Windows 10 version? If so do that and reboot, then try this launcher install again please.

If still no good, do you see Paradox Launcher v2 in the Windows Control Panel to uninstall? If so does it uninstall cleanly, or get an error doing that? You can use this tool to remove it properly:

Fix problems that block programs from being installed or removed

The Program Install and Uninstall troubleshooter helps you automatically repair issues when you're blocked from installing or removing programs. support.microsoft.com