With some banging away at things, I managed to create a workaround by manually generating and uploading favicons using realfavicongenerator.net (which fixed the iOS issue) and then by manually adding code for Twitter cards using their support documentation here: https://developer.twitter.com/en/docs/twitter-for-websites/cards/overview/abouts-cards
It’s worth noting that, before I switched to Tortuga, all of this was being pulled and set up automatically using the “Site Icon” in WordPress. Something about the switch to this theme broke that: does the theme generate its own code for these things, which conflicts? Does it lack support for some browsers? Whatever the case, it might be best to see if it can be resolved in the future, especially given the long turn-around for support here (nearly three weeks on, no response).