What is causing the "search engine doesn't exist" error?

What is causing the "search engine doesn't exist" error?

You might come across the following error in system logs when Live Search is used. The "elasticsearch" part can also be slightly different in various cases, for example, "elasticsearch7" or "opensearch" can be mentioned instead:
report.ERROR: elasticsearch search engine doesn't exist. Falling back to livesearch

It means that, while Live Search is installed and configured in the backend of a Magento instance, the core_config_data table in the database that stores all the configuration info has some other search engine mentioned in it. This happens if backend configs were overridden by hardcoding one of the core Magento files:
app/etc/env.php
app/etc/config.php

To get rid of the error, it is recommended to update the catalog/search/engine record in the core_config_data table and replace irrelevant search engine name with "livesearch".