So for whatever reason, if you fall to your death because no one pulled you up AKA while grabbing a ledge, you die and the bot who is supposed to replace you also dies.

Also the game ends if you don't have a "bot plays the game" script.

I can see the first thing you mention happening.
For the other thing though,the game doesn't end on my end when I die and I properly take over a bot. I do set sb_all_bot_game 1 cvar when dying to prevent the game ending so I assume the issue is caused by some other script

For the second part, it was an issue caused by another script so no need to worry about that.

Which addon was causing the issue ?

Not an addon, but one of my own personal scripts, which I haven't made public.

This mod conflicts with yours @Nescius.
https://psteamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=2608563050

Taking over bots will cause items you had before the take over to levitate below you. Pistols display negative bullets where you can't shoot with them.