I don't think DanzÅ trusted ANYBODY at that point, except maybe Hiruzen. And he trusted him being too soft-hearted to do what was necessary.
Shisui approaching Naruto would actually change a number of things: IIRC Iruka didn't actually soften\teach to Naruto until something like Naruto's last two years or so of Academy. Shisui died after Sasuke's first year.
That means Naruto would get somebody years earlier. Which, in turn, might mean he'd be LESS attention-starved, resulting in him being less of a troublemaker in class and possibly actually paying attention(more often at very least), as well as having a better reputation(I still think at some point people had it against Naruto because he was a little piece of shit, not caring at all about Ninetails).
Getting closer to Shisui automatically means he gets to know Itachi, Sasuke and most important Mikoto(who is not going to let her chance to help Kushina's son: the anime did state she did try when he was a toddler but was stopped, or I'm making up things?). Fugaku might be against it because he doesn't want to bring even more attention to the clan, though, but Sasuke and Naruto ARE schoolmates, so...
It also means lotta people is rushing to the Hokage to tell him those pesky Uchihas are trying to take control of Naruto for a second attack on the village like the one 8 years before.
Which is probably a GOOD THING for the Uchiha clan: it's going to bring all the seeping distrust on the surface, right on the nose of the Hokage for him to see the Uchiha clan wasn't exaggerating much when lamenting of being discriminated. Especially because he trust Shisui and he trust Itachi who confirms Shisui is acting on his own because he was sad for Naruto.