One set of sites (under the same account) that were experiencing this problem were just migrated to a new server (the migration was already planned, had nothing to do with this issue). That seems to have resolved the “Missing or not expected HTTP response headers problem.”
Although I am now able to run backups, a different problem has recurred. When I was last able to run complete backups, they had this warning:
WARNING: mysqli::real_connect(): Headers and client library minor version mismatch. Headers:100020 Library:100106
My host had told me this would be resolved by the migration, but it wasn’t. It turned out to be related to my host using legacy MySql drivers, and was resolved by switching to the native drivers. This may be not be related to the “Missing or not expected HTTP response headers” problem, but I mention it in case it is helpful to someone else.
The other installation on which I encountered the “Missing or not expected HTTP response headers” problem is also scheduled to be migrated this week. The CPU allocation on the server where that installation is currently hosted is heavily restricted, and the server is running PHP and MySQL versions that are quite old. I will post again when the installation has been moved to a server that is a little closer to adequate for a WordPress installation, and update whether that resolved the problem.