This is reaching a lot, if they had merged the show would've made it very obvious and if the BoS just took in the soldiers instead they wouldn't have enough influence to change the BoS at all.

The changes are simply the results of the way Maxson led the BoS in Fallout 4.

It's an interesting theory, but I don't believe it happened. The Brotherhood and Legion's ideologies are too radically different to ever consider working together. The only thing they really have in common is that they're both enemies with the NCR.

Each Elder is different in the way they want to lead

No, simply.

The BoS has always had both latin and religious undertones. They're literally a knightly order.