Breaking Into a HF (PB -> MBA -> AM -> HF) Feasible Career Trajectory?

I am an incoming PB first-year analyst at a BB (think GS or JPM). In three years I will likely move into an associate role within the firm. My passion however is investments with the goal of entering AM, and with enough time and experience, using my experience in AM as a stepping stone into a HF. I intend on pursuing an MBA to break into this industry mainly for the connections, brand-name recognition, retooling, and the optionality a top-tier MBA would grant me as an applicant which I currently do not have with my present alumni base from my current institution.

Background:

I am currently a senior at a non-target undergrad school with a decently strong Cum GPA (~3.85) and substantial leadership and finance-focused extracurriculars/involvement relative to the school I attended. While working in PB, I intend on pursuing the CFA and hope to have lvl 1 and 2 completed by the three-year mark.

Questions:

Is the above career trajectory/plan a feasible one or hot garbage, what is wrong with it? Am I hopelessly locked out of a HF or AM role due to my entry into the industry being PB/PWM?

When should I begin the application process for an MBA, three years into my role in PB or perhaps six? (I have heard six years tends to be the upper-bounds of when people jump into an MBA.)

Is it even worthwhile to pursue an MBA? Should I focus exclusively on the CFA instead and other passion projects to round me out as a candidate to break into AM and then a HF?

Is it possible to move directly into a HF out of an MBA and completely sidestep the intermediate AM portion of my plan?

Would it be possible to lateral into an AM role within my current BB instead of using a CFA/MBA to do so later on?

Thanks!:

Just another monkey amongst the hoard of other applicants who could not break into IB or AM immediately out of undergrad and still hungry to do so. Please feel free to pick apart the above any way you wish any feedback helps. Thanks in advance for any advice!