How would I get all timespent sum by an author through the Rest-API?
Instead of trying to use the time-spent field, perhaps you use the worklog field instead. You can call the REST API endpoint of GET /rest/api/2/issue/{issueIdOrKey}/worklog. This will return all the work log entries and their times.
Also, please refer to this similar issue, https://community.atlassian.com/t5/Jira-questions/Re-Jira-TimeSpent-query-for-the-users-who-logs-hours/qaq-p/1825128/comment-id/504005#M504005 might be of interest,
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