While reinstalling this game again after a few years off, I ran into a problem like others with running the x64 version.
I couldn't find a direct answer as to what to do to fix this problem and still use the x64 version.

Relatively quickly I found a solution by analyzing the event log in the operating system. The problem is with the OpenSSL libraries.
Stonehearth is using these libraries in version 1.0.2h. Just update the libraries in the "/x64" subdirectory and everything works fine again. I verified it on Windows 11 with OpenSSL updated to version 1.0.2i.
Here is a quick step by step guide what to do.

1) Find and download OpenSSL version 1.0.2i (google the file named "openssl-1.0.2i-x64_86-win64.zip")
2) Rename or delete "libeay32.dll" and "ssleay32.dll" from the "/x64" subfolder of the game (eg. C:/Program Files (x86)/Steam/steamapps/common/Stonehearth/x64)
3) Extract the libraries "libeay32.dll" and "ssleay32.dll" from the archive downloaded in step 1) and copy them to the "/x64" subfolder of the game
4) Done