In answer to your question OP, absolutely there is light skin privilege.
My understanding is that its origins are due to colonialism, slavery and proximity to whiteness.
The colonial masters were all white, and people from the countries of the British Empire were taught that white everything was best, so that was what was aspired to. This legacy remains in any Commonwealth country you visit.
Within slavery, individuals benefitted from their proximity to whiteness. The skin of Black Africans was lightened their skin over generations through rape of Black women by the slave masters. As people became lighter, they were moved from heavier duty field work, to house jobs such as ladies maid and butler.
It's no coincidence that the countries in which shadism, colourism and light skin privilege are most prevalent are those such as the US and Brazil, who were the recipients of the largest numbers of slave ships.
I am a dark skinned Black woman, born and brought up in the UK.from an early age, I'm not sure how, I've always accepted that I would be treated as the least attractive, despite knowing I am not. It helped me focus on my studies during my school days, as all the boys (including the Black ones) wanted blondes and brunettes. Even as young as in primary school when we played kiss chase, I remember nobody would run after me, so from my experience it starts at a very young age.