I know both schools.
There is no harm in preparing for the grammar school alongside submitting to JF. It's definitely wise to keep your options open. In any case, only you know your child. To me it's a spiritual thing. Each school will have a distinct feeling as to which one is right for your son.
I think it's scary to label your children at this age also as to how smart they are. The brain is a muscle which grows with continued effort. It becomes smarter through learning from its mistakes. Moreover, you never quite know when they are going to mature. My ds didn't get into his current top private school with the scholarship that I needed to afford it until after three attempts. He is now in the top cohort of his school while his friend who got in at 10+ has had a warning letter about being able to stay on at sixth form.
Child success is a three way thing. The parent, the school, the child. Where my ds is now, even if another school was scoring higher, if on a spiritual level (sorry for sounding whoo whoo), the schools values and ethos were not in line with my values and my son's character I think with all the will in the world he would not be doing so well there.
With regards to having a tutor, I didn't have the academic ability other parents had and so would not have been able to prepare my son for entry myself. Hiring someone with the ability is much the same as a parent tutoring them. There is also the time factor. Do you have the time or stressed to the hills with a full-time job?
I say all that to say, in my view, alongside the scores, an environment where the child can be their best self is important too.