Ownership vs Beneficiary

My grandmother recently told me she has a “NYLIAC VARIABLE UNIVERSAL LIFE 2000” policy on my father, with me as the tertiary beneficiary (behind her and my grandfather).

Her goal is for me to become the primary beneficiary and take over the payments.

I have a few questions:

Should I take ownership of the policy, or should she stay the owner while making me the sole beneficiary, and I just make the payments for her?

Are there tax implications if ownership is transferred to me?

My stepmother is estranged from the family, and I fear she will try to claim this if she finds out. If she isn’t involved in the policy or payments, does that exclude her from any claim on the benefits? How do I protect this?

Is there anything specific about this kind of policy that I need to know? I don’t fully understand what the cost/benefit for me would be if I take ownership of the policy.

Is there a better option for this policy?

Thanks for any advice!

Policy Info: Death benefit: $698,065 Surrender Value: $197,884 Planned monthly premium: $308