I never saw this snowball effect.
In my last game, defeat on Deity, I had instant DoW from my two Autocracy neighbors (I was on Order) when I reached satellites (went for a SV with Maya).
So bribe them also after Ideologies.
Budweiser tips help too. I never had a DoW if I was at least 5 or 6 th in military in demographics.
did you build them landmarks?
vote up their world ideology: autocracy proposal? at the very worst send them lux/monetary tribute?
Most importantly, make sure your neighbors hate each other (war bribes early on in the game will usually do it) so they don't use "shall we declare war on"... on you.
Most common mistake I've seen is to bribe with no reason (it's pretty obvious that it's a mistake when you spend your resources on just triggering a war); these things can also happen:
1) You created a runaway when one civ is able to achieve complete victory over the other.
2) You've made the world hate the person you bribed, in the worst case, your own aggressive neighbor (say they wiped out a CS) and hence you can never be friends with them (and hence getting a green modifier to help dissuade them from attacking you) without angering the rest of the world, including your other neighbor(s). So now you are being caught between a rock and hard place.
Bribe when:
1) army is close to your borders/spy warns you of impending attack
2) someone far away is running away with a small military next to an aggressive neighbor (although sometimes you don't even need to bribe... the AI will DoW on its on accord)
3) To cancel a civ who has RAs and is doing too well technologically for your taste... (here usually an attack sometimes never happens since the RA is signed with a civ that is not their neighbor) but at least, depending on the cost, you've cancelled their RA.
And try to bribe your neighbor's neighbor, who is not your neighbor
to OP:
Don't watch let's plays if you are only a prince player
you'll get confused and you will never be able to pull off their tricks.
put cities on food focus, watch your happiness (build workers) and everything will be fine.