FFXIV Boot Error - Unable to complete version check.

[Kirika-]

(Some rather aged information instructed doing that in IE, but I don't know how applicable that was to the present state of the game); .

It is not dated nor specific to IE
Because the Windows version launcher uses Microsoft's baked-in Windows webUI and its functions/features to perform the task of checking/patching the game files and to connect and log into the game servers, that system must be operational and compatible in order to do so.
The webUI is ran on the same engine as Internet Explorer / Edge, so it uses the same settings and security profiles as those programs, of which are accessed via Internet Options in Control Panel.
Popular methods of locking down a Windows computer security, or to limit the function of the built in web browser(s) (IE, Edge) to directly effect their ability to change system data, do cause a function loss that the FF14 launcher does need. Also, some trojan and/or backdoor malware will manipulate the security and function of that system to it's own ends, which in turn causes the launcher to loose functionality.
That is why the settings in Control Panel/Internet Options are needed as above to the launcher can function correctly.

Other issues that could have been the cause of your issue could have been your Windows User Account security options, or permission levels set on your Windows account that would restrict the ability of a program to write data to certain files/folder locations.
Windows 10/11's built in Ransomware Protection will also sometimes cause issues with the Launcher's ability to write to certain files/folders.