TO be honest, it doesn't happen in 100% of my games. Some games it doesn't happen at all. But when it does happen, it is 100% annoying. The endless coup murder spamming becomes like a program stuck in a do-while loop that will not end until (it seems like) some arbitrary number of turns have passed.
Went for diplo victory in my last game (successfully). Two turns before the UN vote, five coup attempts - three unsuccessful, the other two both on Jakarta (and another successful coup the next turn after I'd GPed the CS back). I think I had a defending spy in one of the targeted CSes (Prague), and influence probably around 200 or less in all of them. And this was an 8 or 10 civ map.
Somehow I always lose Jakarta - one game when my AI rivals were determined to take out my CSes to stop me getting diplo victory, that was the one they successfully conquered.
It's more of a balance issue IMO in that clearly the AI is getting advantages the player doesn't in their couping abilities, which results in this situation and much frustration.
Except that I simply don't seem to experience that - I'd estimate that coup attempts in my games are successful perhaps 50% of the time, with the AI no more successful than my efforts, and not obviously less successful against AI targets than against my allies (probably less so since I specifically target CSes for maximum influence, which I suspect the AI isn't as good at doing).
And in games where the endless coup murder spam cycle (usually lasts ~30-45 turns in my games before it stops) sets in, the game feels broken. You can work around it, but the constant notifications are such a drag. Not to mention the immense amount of resources you can waste trying to keep up the alliances. Then add in Alex's UA or Austria's ability to coup then DM in a single turn and you have a real hot mess to contend with.
Austria was very helpful in my game - she annexed every CS that wasn't my ally, so I needed fewer and fewer votes to win. Also remember that CSes aren't your only source of votes - in that game I did (just, and with the UN) have enough CSes to win even after the Jakarta coup, but I also had an automatic vote from Songhai (liberated) and a vote from my one game-long friend, Korea, giving me a buffer where I could afford to lose CSes. Similarly in the game where Jakarta was invaded, I (just) won because liberated Elizabeth compensated for the loss of that vote.
And with the levels of influence you need to make an alliance last reliably, losing 0.5 influence a turn rather than 1 from Alex's ability is trivial. He doesn't gain any additional influence from his ability, just loses/recovers it more quickly. In G&K it's a basically useless ability.
I get that you don't see it often in your games, but as you noted, many others do. So maybe instead of asking them what they do wrong, you could tell us what you do right? I'd love to be able to reliably avoid this, as I'm sure others will as well.
This is the thing - I've described my playstyle with CSes a number of times on other threads, and I know people who've read them (yourself included) still have these issues. Maybe I should try sharing a saved game - how do I do that?
I'd also like to say that a coup cool-down would help with this situation. It would have to be coupled with a period of time where no one can buy influence, otherwise a player could instantly overcome a coup and the coup[er] couldn't do anything about it.
I used to agree with that, but since then (and following several games where I have had astronomical levels of influence that the AI has no possible way to counter except a coup) I've realised that coups are important as a counter to other players' influence, and need to be available "on call". I now very rarely use coups myself - too risky, too little gain, and the influence you get from them is low so you'll tend to be at greater risk from countercoups (which is probably why three successive coups in Jakarta succeeded, but none against any of my other allies).