Earlier quoted context omitted.
The answer to "job-hopper stigma" and any and all other competence-trigger hurdles is simple. Sell the benefit, not the feature. In other words, change the conversation from "who you are" to "what you can do for them". A track record of accomplishments does a lot more to convince a stake-holder that you know what you're doing than a list of previously-held positions. If all you have is a list of previously-held posit…
Well. Conveying competence is, indeed, a valuable skill. But the most it can do for you is to get you a seat at the table. What will you do once you've got the job, and it comes time to play the game? And, while we're at it, why assume that the only possible reward for playing the game is the opportunity to keep playing the game? Isn't it possible that, in playing the game with sufficient skill and artistry, you can…
Do your job? I'm having trouble understanding what you're getting at. If you don't like your work environment, then leave. Just don't take six months to figure that out. But really, you should know whether you would like working there before you take the job. It's not that hard to take a few of their current employees out for coffee and ask them what it's like to work there.
Not every job has the insane level of politicking described. Just find a place where you fit in. It exists.