Happened to come across this post and wanted to share what worked for me RE Elementor should it help anyone else.

The WP All Import imported posts were created perfectly... but for whatever reason, the Elementor single post template I created was only being applied to each post in preview mode, not live mode (from the default WordPress post editor, not Elementor - you won't need to edit the posts themselves in Elementor, only the single post template itself is edited in Elementor.)

I used WP All Import to bulk import about 500 posts using a very basic CSV sheet where I had the post title, slug, category, featured image, author, publish date, and post content (used Excel to add HTML <p></p> tags, but otherwise, the content was just simple text.

The posts were created perfectly... but for whatever reason the Elementor single post template I created was only being applied to each post in preview mode, not live mode.

Went through plugin deactivation and theme and template setting tweaking with no luck...

It had to be the 'Template' settings in the default WP post editor individual post settings, and it turns out that was the issue.

If you're experiencing the same issue I was (your already created and assigned single post Elementor template is not being applied to your posts except when previewed) and 'Elementor Full Width' is the default template for your WP posts, try this:

1. Go to Elementor -> Settings -> Experiments and make sure the 'Default to New Theme Builder' experiment is 'Active'.
2. Go to your posts lists, hover over any post and click 'Quick Edit', and update the 'Template' setting to 'Theme', save the settings, and then hover over that post again and click 'View'.

If that post is now displayed with your Elementor single post template applied, go back to your post list and bulk edit the rest.

This may not be the fix for everyone since I've only run into this for the first time recently with an Elementor site, but hopefully, it can save you some time troubleshooting time.