Well, for one thing, Jack is more in line with a Vanguard than an Adept in both gameplay and cutscenes, whatever her skill evolutions might say. And technically Shepard can pull off some similar Biotic stuff to Jack as a Vanguard in ME3 (combining the Biotic Charge with Heavy Melee's biotic variant makes for a very badass feeling). It's worth mentioning that Shepard's increase in biotic power in ME2/ME3 came about because Cerberus replaced his/her biotic implant with a newer model. The L5x implant that Jack makes as her unique upgrade is the Adept upgrade and not the Vanguard upgrade, which might actually mean Jack isn't operating at her fullest potential, since the features of L5x as described are more tailored to Adept abilities like Singularity.
"But the lore itself was a mess. Kaidan was supposed to be first generation human biotics (as they are mostly accidentals) and yet Miranda is older than Kaidan and was genetically engineered to have biotics.. like how on earth her father know how to do that especially when humanity don't even know how biotics came to be."
Humanity knew enough about biotics. They were just stubborn about getting help learning how to use them. The actual information about what caused biotics would have been a quick Extranet wikipedia search away once they were invited to join the Council races.