As I work in a typical enterprise environment I have the same problem as this recent post Installing Extension stuck. I posted this there but am raising it as a separate issue for better visibility.
After getting the above I also looked in the Windows Event Log for AppLocker and found:
%OSDRIVE%\USERS\xxxxx\.VSCODE\EXTENSIONS\ORACLE.SQL-DEVELOPER-23.4.0-WIN32-X64\DBTOOLS\JDK\BIN\JAVA.EXE was prevented from running.
Which is going to be another barrier you'll face in typical enterprise environments: Executable rules in AppLocker
I have JDK 8 installed for development purposes and perhaps the extension can have a setting allowing me to configure the JDK Path for it to use? If the extension requires a newer version such as JDK 21 I can easily install that under C:\Program Files for it to use.