Me! me! let me be first - then you can all shoot me down later.
First of all, 'this intimacy' is most definitely NOT restricted to sex. The "amory" part is exactly the point. Multiple sex partners is one thing, more than one love is another thing entirely, although not unnaturally one tends to like to have sex with one's loves.
As to same rights, for me a lot of the poly mindset is freeing oneself from the idea that you have rights per se over another human being. It's about letting them be free to love as they feel fit.
That said, it does seem that many people find that 'other' relationships can take priority over 'original' ones. People sometimes use terms like 'primary' and 'secondary' but for me these don't accurately reflect how I feel, and give the wrong impression. My 'secondary' is just as dear to me as my 'primary'.
Time is always a problem. There are blessed groupings (Vs and Ws) that manage to live together or at least close, but I seem to only manage to attract partners several hundred miles away (bad pheremones??!) and struggle to balance time. On top of that I value 'me time'. So any of my partners tend to feel slotted into spare moments. I wish I didn't have to work!
For what it's worth I have got to a place where I try to let people and relationships be and become what they naturally will. Obviously in this I will and do have my disappointments.
I used to think that to not be damaged personally by such disappointments, one would have to be hard and non-loving. My experience is different. You can indeed keep yourself warm and loving but retain your own sense of self-worth and control of your own happiness. Try to enjoy the pleasure your partner has in his other partners. Getting to know them reportedly helps but as you can imagine this is hard work for me!
Anyway, welcome aboard! It's not an easy ride in the current social situation, but for me the feeling of rightness and alignment with my own mores overrides everything else.
(let the flames begin!)