Re: Why 'Let Rin die'?
1. Perhaps Kishi didn't want to give away the plot twist that Kakashi actually killed Rin
2. Obito said "Like hell I'll acknowledge that!" after Rin's death, so maybe he's in denial that Kakashi killed her. Hence the "let"...
3. Obito already knows what's happened, due to the following;
- He could've asked Zetsu who saw the whole thing,
- He says he doesn't care, but he did go to the Hidden Mist village and cause quite some problems there. It'd be a bit strange if he never found out why people went after Rin during his reign as Mizukage. Perhaps him telling Kisame that "People carry out ugly, nasty missions for the sake of their village and country, even missions of betrayal... This world is full of lies and deceit" is a hint of him knowing the reason and Rin's secret.
- He kinda meant Kakashi when he said "Heroes making pitiful excuses in front of graves", so he might have already overheard the reason.