I don't think that Rasputin was inherently evil. I think that he just took what was offered.
As each section of society began to recognize him, he just let them. I don't think he went to St. Petersburg to become attached to the Imperial Family. That, too, just sort of happened.
If I were to use an old phrase from the 1960s, I would say that he would "go with the flow".
If opportunity was offered and it appealed to him, he took it. It has been said that he kept none of the money nor many of the gifts that he was given. He would just as likely take a gift and then give it to the next person who came through the door. That is what made him seem "holy". His lack of pretension.
I think most of the time he dealt honestly with the Imperial Family and the Empress. I also believe that his taste for the baser enjoyments of life such as wine, women and song were too hard for him to hide. He was lucky that Alexandra already had preconceived notions of how a "peasant" or "holy starets" should act and looked no further. She refused to believe what was said about him because she didn't like the people who said it.
He did have some kind of ability to "help" Alexei. I don't know if the end result justified the means. But obviously Alexandra was willing to take his "help" and over look the problems associated with receiving his help.
People always ask, "If you had a child with an incurable illness, wouldn't you go to any extreme to try to help him or find a cure?" I think I would go to extremes, but I don't think I would turn a blind eye to the information being collected by reputable sources and continue to have that person in my home as Alexandra did.
Here is an out of this world example. If Charles Manson told you that he had a cure for your child but reputable sources told you that he was running a cult and indoctrinating young people into it with absolutely no worldly good coming from this cult except that Manson had a great "s*x" life, and could get a lot of drugs (which he would be willing to share with his followers) would you want him in your home? Would you let your daughters near him? Would you allow him access to the nursery?
That is pretty much what Rasputin did. He had his cult followers and a great "s*x" life. He lived freely without working for what he needed because everyone was only too willing to give it to him. If he wanted food or wine or drugs, he could get them and freely share them but he was most interested in "sharing himself" with them. Just as Manson did.
In our world, Rasputin would have been the kind of person who "falls into a pile of s**t and comes up covered with diamonds".
But in the end, his baser appetites are what led to his death and also his own conviction, by the end of 1916, that Nicholas and Alexandra would protect him as they had been doing for a very long time.
A man with some good sense as in his advice to Nicholas not to go into World War I, but a humanitarian - I don't see it.