I did found out that full text index is done using a Job. If I’m correct, on install without job management available, this is an issue.
Edit: fixed this by finally adding the synchronous strategy to my local.config.php:
'service_manager' => [
'aliases' => [
'Omeka\Job\DispatchStrategy' => 'Omeka\Job\DispatchStrategy\Synchronous',
],
],
Then headed to the settings page and enabled the full text index option. The job ended successfully and now I can search normally.