All right. Here is the answer from Screencastify support:
Thank you for reaching out and I'm happy to help! We have been made aware that there are some issues with the extension downloads on new Macbook Airs and are looking into ways to fix this issue.
In the meantime, following these steps should enable you to download the extension:
1. Uninstall Google Chrome optimized for the M1 chip
2. Install Google Chrome for intel chips
3. Install the Record extension
4. Uninstall Google Chrome
5. Install Google Chrome optimized for M1 chip
When I installed the Chrome version for intel chips, it already remembered I had an account and the extension icon was there. As I can't even figure out how to uninstall this version, and it seems to be working fine, I'm leaving well enough alone...