It is odd that a stronger bond was not forged between such a young John and Mrs. Dashwood; stronger could have gone either way—towards affection or towards dislike IMO. I agree that John’s being sent away to school may be one reason for the missing bond. I think John himself is also part of the equation. In Chapter 1 John is described as rather cold hearted, and rather selfish and a person who does the right thing for proprieties sake and not necessarily because it is the right (morale) thing to do. John’s nature is sort of the opposite of a person with sensibilities like Mrs. Dashwood—perhaps they clashed at times. After the death of her husband she indulges her sensibilities to an extreme which places a burden on Elinor who is also grieving—this is a bit insensitive of her to say the least. I can imagine that if she was as Sensible of her Sensibilities when John was young as she is when the novel opens that it is not completely unreasonable to think that she may have been caught up in them from time to time and that might have made her seem unbending and unreasonable to a young boy who cannot appreciate or understand her despite her normal kind and loving disposition. ;D

it was secured, in such a way, as to leave to himself no power of providing for those who were most dear to him, and who most needed a provision.

This quote does seem to exclude John as being most dear to Henry Dashwood because those most dear to him are also those who most needed a provision and John does not need provision at all for he has been well taken care of by his fortune from his mother and his inheritance from his father is secure. If Henry D had more affection for his daughters it does not mean he had no affection for his son or favored his daughters in any overt way. I think the age difference between John and his sisters would make competition between them rare especially if John was away at school for long periods of time and then began his adult life on his own while the girls were still young. Of course the argument could be made that being sent away to school, no matter how common a practice, is still all a child needs to be resentful of the family remaining in the home. If his father cared more for his second family than John I am not sure he felt it but I do think the fact that H. Dashwood felt he had to secure a promise from John for the security of his wife and daughters shows that he was close enough to John to be aware of his selfish nature but he did not know him well enough to worry he could be talked out of his promise.

In Chapter 1 JA is careful to say that John is emotionally a cold-fish and different from the rest of the family but she does not thow any of the common troubles found in many step-family relationships such as jealously, resentment and favoritism of the children of the other marriage to the mix. Before his father dies, John seems to me to be neutral towards his step-family only waiting to take a clue from someone else on how to act. He says the right things but I do not feel there is any affection or particular resentment behind his words to make the sentiments real or false. When his father asks him to help them John is affected enough to plan what seems a reasonable amount of assistance but is easily persuaded by Fanny to do less and less until it is practically nothing. Since Fanny is a "strong caricature of himself; more narrow-minded and selfish” IMO she is the voice of his cold heart and thus is the stronger influence.

Your father thought only of them . And I must say this: that you owe no particular gratitude to him, nor attention to his wishes, for we very well know that if he could, he would have left almost everything in the world to them." This argument was irresistible. It gave to his intentions whatever of decision was wanting before; and he finally resolved, that it would be absolutely unnecessary, if not highly indecorous, to do more for the widow and children of his father, than such kind of neighbourly acts as his own wife pointed out. (Chapter 2)