Apart from what you said, I think there's a big problem with the way he coaches his players, and at present though there are a few glimpses of improvement I don't really think there've been enough to conclude that he has improved in that regard. We'll see as the season goes on.
The thing that I see with Jose is - and this is coming from a fans perspective is that ultimately what a lot fans wanted have proven to be right.
Whilst he said that he was playing the likes of martial or Shaw due to fans wanting it - it has ultimately been proven successful right?
The players that we fans want the team to correlate around have been more successful than Jose's choices. I ultimately see the tactics alpt of the fans talk about better than what Jose puts out on the pitch - except for Jose's match winning capability of changing a few last minute tactics that he adapts to get the best out of his opposition - that is why he is historically been a great manager; because he can outwit the opposition well.
Lukaku was the main guy our attacks revolved around & now that seems to be martial. Players like felliani are great at our plan B rather than plan A. Players like Shaw have been great. Players like Rashford have been ruined in terms of how he approached football by playing him on the wings in my opinion & even to this day fans or ex players like ri0 say that Rashford is meant to be our striker - at least over Lukaku. The list can go on -
However just because Jose is trying to sort out the mistakes he previously made doesn't make him the best man for the Job does it? I'd still love to see what a manager who is fully flexible to adapt his tactics towards his players can do at United. To this day I think 352 with a front 3 of Sanchez, Martial & Rashford is one of the best attacks in the PL - do I see Jose using it though? Doubt it.
Ultimately - if Jose is here then the players that would allow him to be the best man for the job is players like Lukaku, players like perisic & Willian supplying him, players like felliani & Matic having vital & important roles for the club.
On the other hand - is Jose the man to unlock martial, Rashford, Sanchez, pogba, pereira,, shaws hell even players like chong & greenwood's potential - I simply don't see why that is the case.
The whole Woodward vs Jose thing is clearly because both of them had two different paths to take United. One wouldn't be too worried about playing martial as a bit part player whilst another was adamant that he should be one of the main players at the club. That resonantes with alpt of the decisions I have seen so far in this 2. 5 years.
If Jose was the best man for this type of player & this type of club - he would have adapted his tactics the moment he came in his first year - however we see it in his 3rd year after his tactics has ultimately failed and or been disagreed upon.