It looks like I can't edit the timeSpent field in the issue object, neither via create issue or edit issue (https://docs.atlassian.com/software/jira/docs/api/REST/7.6.1/?_ga=2.80820181.351714748.1519344052-55782724.1518460860#api/2/issue-editIssue) endpoint. Is there some other endpoint to do this, or is it strictly the summation of timeSpent of associated worklogs?
It is the summation of associated worklogs. So, you will have to use the "Add worklog" API.
POST /rest/api/2/issue/{issueIdOrKey}/worklog
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