First of all, your statement about adding natural weapons as secondary attacks after a flurry of blows is incorrect about unchained monks. The original monk got an errata-nerf that prevented this, but the unchained monk does not have that line. This also makes the rules more consistent, and is how things worked in D&D 3.5. It never caused problems.
But as for using a pair of primary claws as basically one attack, and then gaining a bonus attack from flurry of blows, that is ambiguous. Normally, when you use primary natural weapons, that is in place of iterative attacks. But flurry of blows bonus attacks are just that, bonus attacks, not iterative attacks.
Personally, I would allow it. Monks are one of the weakest classes in Pathfinder (see here), and in general gaining natural weapons also costs you something, so it seems entirely safe to me to combine the two. This is how things worked in 3.5, and it never caused problems (monks still sucked). The unchained monk is a better, but still not good, and natural weapons are usually overpriced, so if anything I would be worried that combining the two cost too much and was underpowered.
On the other hand, it seems reasonable to me to interpret Paizo’s statements about flurry of blows as indicating that in their mind, absolutely nothing ever should allow a monk to attack more times during a flurry of blows than it says on the table. This doesn’t make much sense to me (does that include haste? I would assume not, because why would it—but then what is the difference here?). Paizo has made a number of statements that don’t actually make sense with the rules as written trying to prevent various ways of gaining more attacks (see the entire “flurry of blows is two-weapon fighting” fiasco). So if you are unduly impressed with Paizo’s design skills and want to absolutely follow their ideas on game balance, then you probably would not want to allow it.
And the reason we have this situation is because, basically, the natural weapon rules of D&D 3.5 are a mess, and Paizo has only made things worse with the ways they have tried to edit the rules. It seems to me that they are dead-set on trying to claim that the 3.5 rules always worked the way they say the Pathfinder rules work, instead of admitting the 3.5 rules didn’t work that way (they didn’t), and then just changing the rules for their system. So instead of clear rules that say how things work, we have unclear rules that try to take 3.5 rules and imply that they always said what Paizo wants them to have said (even though they don’t). I do not know why this is the approach they took (but then, it isn’t that unusual for Paizo).
Monks are a bad class. They are very poorly designed, and they function very poorly in practice at many, many tables. Their abilities are an antisynergistic hodgepodge of tightly-limited effects and weak approaches to combat. The unchained monk is better, the qinggong monk is better, but ultimately they still leave a lot to be desired. Natural weapons are likewise a mess, and often cause problems of their own. My recommendation is really to just avoid both. A wise, mystical, bestial fighter could be a druid, and be a very similar character, just better, than trying to do the same with monk.