Thanks for all your advice. Here's a report on my experiences for anybody that may have a similar question. I demo'd Zulip with Jitsi, Jami, and Nextcloud Talk today.
I've actually used Signal desktop for years, and love it, but don't really want to mix work with personal, and if I didn't already use it, I'd squirm about the required mobile app on my personal phone. Telegram might be a little easier that way-- I don't think you have to have an app to make an account, just a phone number.
I liked Zulip, but I didn't like Jitsi. It required a google account or a github account to host a meeting, it ran in browser instead of embedded in the Zulip app somehow, and the Jitsi desktop app seemed to be from 2003.
Jami was ok. I was able to set it up pretty easily, but I didn't know if others in the org can handle that it's P2P so you can't leave messages for others off hours. Also there seemed to be a lot of complaints about its reliability in its own forum.
NextCloud talk on the free servers didn't really work. I could get voice and text, but screen sharing just errored. I think I'd have to set up a TURN server or something like that. So if that requires hosting anyway, might as well do Matrix. Also the free servers were really slow.
So next test for me is Matrix. Is there a way I can try it out for free to see if I can recommend it to the rest of the company? Without spending hours on it? I've probably wasted more company time on this project than I would've saved in subscriptions.
Yeah, I care about privacy and I know this community is active, so I posted here. But you're right, this isn't a very privacy focused post. Do you guys have any recommendations for active Lemmy communities that would be a better fit? Piracy@db0 would be a trip I'm sure, but at least it's active. I hate when I've posted and nobody responds.