Seems to me like if you are getting 1 ES every second, yet not enough errors in total to reduce SNRM enough to disconnect the line, a possibility is that you might be getting repeated very short bursts of interference at about 1 second intervals.
Taking a wild guess, one thing that I think might fit that pattern would be farmers’ electric fences. Is that a possible cause, where you live?
Even if not electric fences, maybe still worth wandering around with an AM radio. A pattern of ‘clicks’ at about one second intervals might prove my theory, then it’d be a case of figuring out what’s causing them.
Just a thought.