Jira next-gen projects do not allow to add any additional information to the issue cards yet. It is Atlassian Jira limitation that might be removed in the future.
Open board settings
Open "Card layout" page and add "Checklist Progress" or "Checklist Progress %" field to the card layout.
Enjoy the checklist progress visible on issue cards
Checklist progress in Search issues view
Open "Search" view, click the "Columns" dropdown button and add "Checklist Progress" or "Checklist Progress %" field to the list
Enjoy the checklist progress visible on the search list