Blocks can be disabled though, so perhaps a plugin or even your theme has done so?
Please attempt to deactivate all plugins and switch to the default Twenty Twenty-Three theme. If the problem goes away, re-activate them one by one to identify the source of the problem.
If you can install plugins, install Health Check. On the troubleshooting tab, you can click the button to deactivate all plugins and change the theme for you while you’re still logged in without affecting normal visitors to your site.
At the risk of being completely and publicly clueless… I went into Troubleshooting mode, disabling all plugins and activating the “default” theme, and still no shortcode option…
Open the block editor and click on the 3 dots at the top right, then on “Preferences” and “Blocks”. Look for “Shortcode” in the list. If it is deactivated there, activate it here.