Koeman was before the Silva saga, and the questionaire saga. And I don't remember Everton fans ever viscously hounding out any managers, the longevity of Moyes and the good send off you gave him would have given Everton a good rep as a great club to work at.
But perceptions can change very quickly, and with all due respect you aren't Chelsea. But Everton still have a good rep of being a 'proper' football club, so I don't think that is something you should throw away too lightly.
If you were going for 2 job interviews, both similar pay, both similar sized outfits.
But at 'company A', they have a good rep for giving you time to settle, giving you space to make mistakes because they realise mistakes are only human.
Or 'company B', where they could potentially leave you high and dry during negotiations, humiliate you, and what's more the customers might be liable to come in and treat like you dirt at the slightest mistakes...
I think you'd probably go for company A.
Everton will always be able to attract a certain calibre of manager, but the most sought after up and coming managers might have similar offers from similar sized clubs, and that's where your rep might come into play.
You've still actually got a good rep I think, negotiations break down all the time, marketing teams make mistakes, but if this Big Sam thing turns nasty, that could burn it into people's memories.