I believe the responsibility on this falls at the feet of the father. For whatever reason he didn’t take his son to shul at an earlier age. Once a child learns to daven in school he should go to shul with his father and daven in shul. Of course every child is different and not every child can be expected to sit through the entire shabbos shul experience. Every father needs to judge for themselves how to work that out. In this case the father started taking his son to shul at 9 and had to bribe him to come. This worked for both of them and they also understood the ground rules and where it would end.
They had an agreement in place and it worked for them. There is nothing wrong as long as they both understood how it was going to work and how and when it was going to end. I don’t recommend this as the best way to go about this, but in this case since it worked, it worked. On the other hand, the boy could have negated his end of the bargain at the end of the year and pushed for more bribery. Bribery is NOT a good idea in any situation. A reward and consequence system is always a good way to teach responsibility but not when it is used to manipulate one and other. Rules should be very clear cut and should be in place in advance.
In this case I believe that the father started too late with his son and he should have made it clear that it was the son’s responsibility to go to shul at this age. If he did not require him to go to shul at 7 or 8, he should have been training him at that age, that when he reaches the age of 9, he will then be a big boy and he will be responsible to go to shul with Tatty like a big boy. Making it the son’s responsibility would then put him in a position that he is required to do this, that siblings younger than him are NOT, or may not even be allowed to do this, and this is a coming of age thing and something to be proud of like riding a bike, crossing the street, etc. However, with that privilege comes the responsibility and commitment. That means there is a consequence if you don’t follow through and do what you are supposed to do. This is the total opposite of being bribed.
It is the son’s responsibility to go to shul to daven. This is an important part of growing up and if it is important to both parent and child then parents have to show the importance. If it is NOT then the child should NOT be bribed to do it either. That is NOT good chinuch and that bribe was a little too manipulative. What does davening in shul have to do with having a day off from school? Why would he or should he deserve a day off from school for doing what he is supposed to do? Why not offer him a Hot chocolate, doughnut, bagel, a ride to school after davening or extra time with Tatty each time he goes to shul with Tatty. Not all yeshivas want kids to go to shul with their fathers and would rather have the boys daven together in school. It all depends on what the yeshiva wants. But if the rule is to go to shul, then a child has to be taught and trained to go because it is their responsibility and it is the right thing to do.