How to stop Catalina from contacting Apple servers when executing programs?
Yes, as a security measure, macOS Catalina sends some small info (can be seen by packet size) about unsigned executables to the Apple servers.
A simple  C program as illustrated in ..

Does macOS phone home to Apple&#39;s servers before running an app for the first time?

.. can demonstrate the lag in startup on poor internet connection conditions.

https://lapcatsoftware.com/articles/catalina-executables.html

The article has detailed observations, tests and commentary on the process.
You can block the syspolicyd process from connecting to the internet, and it won't hinder its normal functioning locally, using a firewall.
I use LuLu, since it's free. A lot of people use Little Snitch.

https://github.com/objective-see/LuLu
https://www.obdev.at/products/littlesnitch/index.html

Many other apps also ping their servers on startup or quitting: Firefox,
Adobe,
Sublime Text etc.
Other threads:

Which free alternatives to Little Snitch can I use to block internet traffic per applications?
How can I get the exact URLs/requests that a program is trying to connect to?
Reasons to prefer Little Snitch over the built-in firewall
Firewall for outgoing connections