Msft makes simple things harder than they should be ... one way to include the content, is to save the list as a template and recreate it on the destination site.
Go to the source list > settings and copy the list guid "list=xxxxxx" from the url.
Append this to the site url, replacing the "XXXXX" with your list guid: /_layouts/15/savetmpl.aspx?List=%7BXXXXX%7D
In the page that opens, fill in list name etc and check "include content".
In the destination site, open /_catalogs/lt/Forms/AllItems.aspx and upload the template there.
Go to add an app and look for the template to create the list.
The pointless Msft roadblocks you'll need to overcome for this to work are:
You'll need to be a site collection admin, and enable custom scripting for this to work, which requires SP admin permission.
To enable custom scripting, open the Sharepoint management shell (download it if you don't have it installed) and enter:
Connect-SPOService -Url https://xxx-admin.sharepoint.com (to authenticate)
Set-SPOSite -Identity https://xxx.sharepoint.com/sites/matt-test -DenyAddAndCustomizePages 0
Maybe an easier option which I sometimes use, is to export the source list's content to excel, and then create the new list from that, though you may have to play with image urls for them to show up. One tip is the excel data has to be formatted as a table.
Sharepoint Designer does indeed still work, and you could try copying a list from one site to another there. I don't recall if that works or not. You can definitely copy items though.