I'd imagine you're right on that front, but hypothetically, International management is probably the only situation in which you could replace the manager even with 0 games to go before the WC and it have negligible detriment to your chances.
Each round of fixtures is like starting a fresh almost anyway, different group of players, there's never any real consistency anyway that you would be sacrificing by a change at the top, and WC/Euros are really the only occasions where the group spends a real sustained period of time together to build fluidity and work on becoming a functioning unit.
When the only consistency you've got is turgid, disjointed, garbage, it's not worth clinging onto merely for the reasoning that the WC is so close.
I saw a quote from him the other day, that he would "block out the criticism" and continue to do his thing, that's fucking mental, just bury your head in the sand then Gary and pretend everything is fine and nothing needs changing, that will definitely help. He's either fucking stupid, or incredibly arrogant, I realise that he's not going to reveal to the press that it's a complete shit show and he doesn't have a clue what to do to sort it out, but the criticism is there for a valid reason, he should've acknowledged it and said that the team will continue to work hard and look for ways to do better.
Of course, words are pretty inconsequential anyway when you go on to get tonked 4-0 at home to Hungary.
But still, you're right, the FA won't rock the boat.