So as you already know, list web parts are not exportable via the UI menus. That option is inconveniently missing. There are a few ways however, to get around this but the option I will describe here does NOT require SharePoint designer. Now, now, quiet down with the cheering and the celebration and follow these simple steps:
- Navigate to the list you want to export as a web part and Edit the page
- Open your browser’s Dev Tools and search the HTML for MSOMenu_Export
- In your list page, open the list web part menu (as if you were going to delete/minimise/export it). This is very important.
- Now you are prompted to Save or Open the .webpart file and voila!