Bumping this ticket. Having the same issue on Drupal 7.91, PHP 7.3, and elysia_cron 7.x-2.9. When running cron via drush (`drush cron`), elysia_cron is triggered and runs successfully, but does not update the "Last run" time as seen in OP's screenshot (it does update the individual jobs' run times, same as OP). When running via web request, the "Last run" time does update correctly; however, drush is preferable for our setup at the moment.
It looks like `elysia_cron_drush_invoke()` simply runs the cron tasks and then calls `elysia_cron_drush_die()`, resulting in `drupal_exit()` getting called and terminating the script. This seems to be intended behavior, so that *only* the elysia_cron job runs. However, exiting early prevents the `variable_set('cron_last', REQUEST_TIME);` line in common.inc from getting called, so the "Last run" time isn't getting reset.
For now, I'll just write and apply a patch, but an official fix would be great.