It's not a matter of time invested - as you say, we have time and we're not really in a rush. As I see it, it's a matter of being respectful to the various parties involved. Rin has only just spoken her very first words in this manner and given her initial reaction, we should probably give her some time to get used to it before we thrust other people, who she doesn't even know, into her head. At least to me, this sort of communication feels a bit more intimate or personal than simply talking on the phone or in person would be, so I'd like to be mindful of that. And Kiku in particular is a woman who is used to a lot of formal boundaries, so outside of an emergency, I think it much better to at least ask her permission before trying to initiate a conversation in her head with a 'stranger'.
Honestly, I think it would be better to simply introduce them in person first, if there's not a pressing reason to do otherwise, though I'm not categorically opposed to doing otherwise if we take it a bit slower and ask the other parties involved ahead of time, rather than assuming on them.
>"Oh! I should tell the others!"
>Let's try for a Confrence Call with the rings! Hello? Testing, Testing. Can everyone hear me?
>So hold off on this for a bit