How to fix "Windows could not search for new update" error?

[Snick]

iko22

I may be mistaken, but I believe Simplix requires Win7 SP1 installed.
Download Simplix UpdatePack 7 - MajorGeeks

May want to have a look at DISM++ from their website. Don't need repetitive restarts as Simplix has if you have enough RAM. Also provides you will problematic update notices in Orange, with descriptions too but doesn't update M$ products such as Office, etc. If you receive an error you need to place wsusscn3.cab in Config folder as is, CAB file.

OP may need to install Windows Update Agent from MS catalog, SP1 is prerequisite for it.
Win Update Agent x6

Snick

 

[Volume Z]

Yes, I knew SHA-1 has not been supported as of August 2019 due to security concerns!

Our issue emanates from the changes of August 2020.

Windows 7: SHA-1 Windows content to be retired August 3, 2020
SHA-2 signing enforcement on Windows 7 is almost here

That's two times the same URL provided by you, and it pertains to Microsoft Defender.

SHA-1 Windows content to be retired August 3, 2020 - Microsoft Tech Community - 1544373

That one makes no specification as to which OS will be impacted and to what extent.