In agreement with the sentiments above, I'd say the continued popularity of Sidney Poitier comes down to dignity.
Poitier maintained his own dignity on screen and so offered audiences of all colors and backgrounds a chance to enact/activate/appreciate their own humanity in regards to the presence of a person of his skin tone in film.
I thnk that Poitier's acceptance reveals some very interesting dynamics about audiences. In general, though, I think that it's hard to find an actor who was up against so much and was able to accomplish so much more.
At a time when Black actors were stereotyped when featured on screen, Poitier was a breath of fresh air to cinema. He did not portray the stereotypical "Black" character on screen. He did not "shuck and jive." He did not play the dimwitted sidekick. He was not the minstrelized Jim Crow or Zip Coon. Instead, he portrayed characters with a range of emotions. His surface of elegant dignity was matched with a collection of emotions percolating under this surface. His ability to bring out multiple facets in his on- screen depictions made him beloved by all audiences.
For many members of White audiences, it was the first time they encountered a Black hero on screen. It was the first time they were able to look at a Black man with adulation, praise, and a sense of envy. The elegance that he displayed challenged White audiences and everything, from how he looked to how he spoke to how he carried himself with a seemingly effortless grace made him appealing to White audiences. Poitier knew this. He understood that his presence on scree might be one of the first times that White audiences had to "respect" a Black person. In a major sense, this transformation of audience members helped to enhance the Civil Rights dialogue that gripped America at the time of his acting.
For Black audiences, Poitier was complex. He represented what Black actors could do in terms of being in a leading role and having significant impact on storylines and plot developments. He also represented the fundamental idea that Black actors did not have to check their dignity at the door in order to get a role or to be respected. As the Civil Rights focus turned to art, Poitier was criticized for being "too respectable" or "Uncle Tom." Yet, this criticism was met with equal fervor by Poitier, himself, who argued that continual exposure to depictions of prostitutes, pimps, and street hustlers can have a detrimental impact on Black youth, limiting their own dreams and their visions of self. At the time, this criticism was dismissed. Yet, this same argument is given today in regards to rap music and the depictions of life in rap videos and is seen with credibility. Over time, Black audiences have come to respect Poitier as living proof that "the fight goes on, the cause endures, and the dream never dies." It is for this reason that Poitier is so beloved as an actor and a role model for all.