It looks like all of their tweets are just links to items on the news room portion of their website. If you click around a little there, you'll see that they do have RSS feeds:
So it looks like they're already doing exactly what you suggest: they post official announcements on their website which you can subscribe to using the standard way to do that (RSS), and they also rebroadcast on Twitter by linking back to the original source. What should they be doing differently? Periodically tweet reminders that you can subscribe directly to their RSS feeds? Stop posting to Twitter at all and leave only a message that you can find official news on their website?
https://www.sec.gov/page/news
https://www.sec.gov/about/sec-rss
So it looks like they're already doing exactly what you suggest: they post official announcements on their website which you can subscribe to using the standard way to do that (RSS), and they also rebroadcast on Twitter by linking back to the original source. What should they be doing differently? Periodically tweet reminders that you can subscribe directly to their RSS feeds? Stop posting to Twitter at all and leave only a message that you can find official news on their website?