I actually agree with your underlying point here, that it would be good if the board didn't routinely just react to fan sentiment and make their decisions based on that. In this instance though, I do think the feelings of the fans are a key factor. The club has been divided and drifting for a good few years now, and for the first time in ages it feels like people are united and buying in to something. Throwing that away could set us back again, and that is really the last thing we need. For that reason Lampard should be given every opportunity to build something. At some point he'll need to do more than just be likeable and 'get us' but for now it's actually a pretty important thing.
Fans will turn without doubt you could already see it after Newcastle, i dont necessarily agree with it, but we would be fooling ourselves to believe people wont start getting antsy and beginning calling for change if we go 3-5 consecutive losses. The Palace result and performance cut it off at the pass.
Familiarity also breeds contempt, the longer he is here, the more people will be open to a change, equally expectations will grow, we might be happy being safe by March/April (hopefully), but expectations will grow again next season. Thats the game unfortunately and managers know the score, rightly or wrongly.
Personally i think we are three year job in my opinion to be challenging for 8th+ - that needs consistency and stability, but whether we have the leaders at the club in terms of board and owners or not, not to cave to fan sentiment - im not sure we do - we have made wrong decisions in the past based on fan public opinion, the club are very pliable to it in my opinion, which is no way to run a football club, as we have seen.