To give IndieGamerChick some credit, looking at her tweets regarding the subject again, I think she mainly seems to talking about the NES era Rare that Miyamoto disliked. She did say that he "thought of them making some of the worst NES games".
Honestly...if I were Miyamoto and saw Rare during the NES era, I probably wouldn't think much of them either. Maybe not "despise" as IGC's sources put it, but...Rare did honestly kind of churn out a lot of garbage before they joined forces with Nintendo. For every Battletoad or Snake Rattle 'N' Roll, there were like, five Taboos or A Nightmare on Elm Streets. Quite a few of the games Rare made under LJN's wing ended up being featured on the Angry Video Game Nerd. Think about that for a moment.
So yeah, I guess it is possible that Miyamoto initially didn't trust or like Rare at first, but after the company proved itself with constant hit after hit when they partnered with Nintendo, he warmed up to them.