AFAICT, base backups running on the replication protocol are always reported as "idle" in pg_stat_activity. This seems to have been an oversight in the "include walsender backends in pg_stat_activity" in 10, which does include it for walsenders in general, just not for the ones sending base backups. (and was then improved on later with the "include all non-standard backends" patch).
Unlike the regular walsender it also has to set it back to IDLE, since you can actually finish a base backup without disconnecting.
PFA a patch that fixes this. I think this is bugfix-for-backpatch, I don't think it has a large risk of breaking things. Thoughts?
Also, in setting this, there is no real way to differentiate between a regular walsender and a basebackup walsender, other than looking at the wait events. They're both listed as walsenders. Should there be? (That might not be as easily backpatchable, so keeping that as a separate one)