In the article that you provide the link to the document clearly states that this WinRE partition change only applies to Windows 11 systems that receive their updates via Windows Update (WU) or Windows Server Update Services (WSUS):
"Microsoft has changed how it updates PCs that run the Windows Recovery Environment (WinRE). WinRE will be updated using the monthly cumulative update. This change only applies to PCs that get updates from Windows Update (WU) and Windows Server Update Services (WSUS). This change starts on June 27, 2023, for the Windows 11, version 22H2 cumulative update. ..."
Are you certain that a change that refers only to Windows 11 is valid for Windows 10?
Furthermore the article states in its second paragraph:
"You will receive the error message, 'Windows Recovery Environment servicing failed.'" I don't believe this to be the same error everyone has reported here. Additionally, as per the article, a system may not have a Recovery partition (the article provides a method for checking for the existence of a recovery partition).
The type of change that the article discusses is not for those without a fair amount of experience and can result in a system that is unusable should any step be missed or not completed properly.
Personally I'm not willing to risk my system trying these steps. If this is why the update fails then the same changes made by Microsoft to handle the issue for Windows 11 need to be provided as part of this update for Windows 10 systems. Currently Windows 10 is still a supported OS, so it is Microsoft's responsibility to correct the issue and create a new update that handles an issue they've created. A solution that provides the support that they've stated they'll be providing until October this year--I think the support ends in 2024 and not 2025; either way the OS is still under full support.