Depends.....
personally, i think there is no such thing as ONE recipe for all women on this topic.
There are to many variables, like sex drive, relationship status, cultural background and personal preferences.
The "i have a 100" screen and pay for drinks and food so she has to initiate" argument is ridiculous, go pay a prostitute to initiate and save the money for the big TV and drinks.
Being needy and pushing for sex every time doesnt do you any good in the long run as it puts her in a position of power, unless this relationship is defined as fvck-buddies.
That said, OPs woman might be of low interest in sex but compliant, she might be compliant for now (that might change), she might be quite enthusiastic about sex but it could be a thing of her upbringing in UA that men initiate, or she just likes to be told what to do..... who knows. Its OPs job to find out on what basis she operates.
Starting in my early 20s, i was in a 5 year relationship with a woman that never initiated from the start. After year 1 i was thirsting for her pvssy but only got it every other week. I took it like the little simp i was, grateful for any action i got for 2 more years. After 3 years i was at a point when i thought fvck that bytch, i dont need her and went cheating on her with OLD.
I stopped initiating/giving her physical attention and soon enough she came around showing sudden interest in me, initiating herself.
While i kept fvcking her when she initiated, i didnt initiate anymore and that drove her nuts to a point where she allmost begged me to fvck her.
Looking back my guess is she cheated on me before i withdrew attention from her. Just when i turned the table and took away her leverage over me (sex), i became interesting again.
At that point though i had lost all feelings for her, the relationship didnt recover and i ended it after i had several affairs during that time.
Of course escalating and initiating with a new woman is crucial but, from that point on i never gave any woman sexual leverage on me anymore and it always payed off in every relationship longer then a couple of months.