Francis could easily have added Red Rackham's treasure to the rest of his fortune but instead he kept it hidden, probably as some kind of financial backup.
I'm not so sure. Admittedly I have no idea of what the law was back then, but would Sir Francis have actually had any legal
claim to the treasure? The treasure itself was ill-gotten gains, so he might have had good reason to keep it secret. Legally it might have belonged to any number of people, perhaps relatives of Red Rackham, or the people he stole it from, or even the Crown. Even if he'd had any legal claim to it there would have been good reasons not to advertise the fact, such as avoiding unwelcome attention from thieves, or from Rackham's friends or relatives, or of any of the pirates.
With that in mind, it makes sense that Sir Francis should leave out the part about him "stealing" the treasure in the detailed account written in his journal. Instead he left a cryptic instruction to his sons to move the mast of their model ships slightly aft, "thus the truth will out". For reasons we'll never know, they didn't do this.
I really like the idea mentioned here that the brothers may have had some kind of falling out. Indeed, the wording on the parchments seems to suggest it; "three brothers joyned, three unicorns in company sailing in the noonday sun will speak", which might not only have been a coded instruction, but also a metaphor for their strained relationship. So, perhaps Sir Francis set the whole thing up with more than his inheritance in mind, such as a way of reconciling his three feuding sons? It's a nice thought.