So if the phone gets a security update for this at the OS level, should we theoretically be safe to use apps with any sort of browser functionality? Like some apps that don't update, or are no longer being maintained, etc
This isn't just a browser vulnerability. It's a vulnerability at a much more fundamental level, which is why it's so critical. It's a vulnerability in how almost every piece of software processes a widely supported image format, so anything that touches images is potentially at risk: browsers, chat or messaging apps, file browsers, or really anything that uses thumbnails or image previews, including some core OS functionality. On the server side, you've got anything that makes thumbnails and previews, too.
We should wait and see whether there are any practical attacks outside the browser context (maybe the malicious code needs to be placed in a web page that displays the malicious image file, or maybe they need to figure out a way to actually put all the malicious code in the image file itself). But the vulnerability itself is in a fundamental library used by a lot more software.