I'm not sure, but from a superficial reading I don't think it's currently possible to execute this ticket as defined.
DM-17566 deprecated overloads of afwMath.BackgroundControl.__init__ which take a style parameter. Per this ticket, they must therefore be removed before release 21.0.0.
However, lsst.meas.algorithms.SubtractBackgroundTask.fitBackground() takes an optional algorithm parameter. I believe that this parameter is only meaningful in the presence of the (deprecated) style parameter to BackgroundControl.__init__. However, this interface is not marked as deprecated, and so can't be removed.
I conclude that we are stuck between a rock and a hard place — we have to drop the style parameter because it has been deprecated, but we can't because it'll break non-deprecated interfaces.
I think, therefore, we have to make a new ticket to deprecated the current interface to [lsst.meas.algorithms.SubtractBackgroundTask.fitBackground() before the 20.0.0 release.
Am I correct? I'd appreciate a sanity check, because it's late and I might have confused myself!