Unfortunately, my Concerto system (2.3.6 Debian packages) won't allow this.
If I subscribe to Weather or RSS on a Dynamic field, I get "X approved items" where X is the number of approved items in the feed.
If I subscribe to Weather or RSS on a Text field, I get "0 approved items" regardless of the approval status of items in the feed.
I can replicate this behavior on other types of feeds too. For example, attempting to post a graphics item to a text feed results in exactly the same experience.
And to me, _this_makes_sense_ because otherwise why have a dynamic feed defined at all if we're just going to use text anyway? (Maybe I feel that way because I prefer to work in Perl, where you'll get warnings about variables defined but not used.)
I understand how to approve items. I've worked out how the RSS feeds use the hidden sub-items to retrieve content, so a single RSS item on an RSS feed will result in "2 approved items." I have a graphics feed in production at CBJ, used by the city clerk. The harbors wants to use the system, but they want weather data and that's what I'm trying to resolve.
So I'll make an educated guess here, and ask if at some point in the past you've edited some part of your Concerto system to allow dynamic content on a text field. If so, just let me know what that is and I'll add it to my notes. Or, there is some undocumented 'trick' that I'm missing. (Like the fact that you must have a background image or CSS in a template to show anything at all. I was confused about that until I ran across that in this group.)
Anyway, here is what I see. If you catch an error that I've made, please point it out. Screen shots to follow.
Below is an example of a test template I built, displayed on a screen:

Each field is subscribed to the same two feeds, one of them an RSS feed the other one plain text.

Here is the RSS_TEST feed. I've been pretty liberal with allowed content types:

If I edit the RSS feed, I see the preview, so I am confident that the actual RSS updates are working:

Here's the test text feed, also pretty liberal with the content types as I've been experimenting:

So let's return to the screen and take a peek at how it looks:
The top feed is dynamic content. It says that I have approved RSS items, but no approved text items. (Which, again, _Just_Makes_Sense_ to me. RSS is dynamic, right?)

The ticker feed says that no RSS is approved, but the text is approved. Unfortunately the ticker doesn't tick, so it's just like a text feed.

And finally, the text field says no RSS approved (because RSS is dynamic content, and goes on a dynamic field?), but my text is approved:

And the screen preview shows this. Notice that the text appears where it shows up as approved, but the Dynamic feed does not ever appear.

And here is the template for the screen so you can see I'm not doing the white-on-white CSS error you suggested earlier:
