Just as a side note, I had the same issue a few days ago (Update available → seems to update but jumps back to available after some time) and solved it by cleaning the build via ESPHome Dashboards and installing via OTA to the device again. This builds the files with the new version.
On one device I had to “install” multiple times due to the connection gets terminated by power saving measurements.
Now they get updated perfectly again via HA update procedures.