Backdrop community pay a lot of attention to UX issues. And it is priceless. As a result Backdrop is several steps ahead of Drupal (and many other CMS) in terms of UX.
I would like to propose new UI button group component for Backdrop. It is select component which act like radio buttons or checkboxes. You could see it many times before all over the internet. It is present in bootstrap css/js framework.

It is perfect solution when you have just few options to choose. Its main advantage that you can see all options immediately without additional clicking. Button group needs one click less than selectbox to provide action.
Here's how it might look on Manage content page. Both options are chosen. It is analog of "- Any -" option.

Here's one option chosen.

Manage user accounts page.

This component can be used widely. It is just a sketch. Styles needs to be discussed.
Recent comments
Hi! The description is still very vague and lacks step-by-step instructions on how to reproduce. It doesn't include the version of Backdrop either, nor a list of contrib modules you are using...
Problems with HTML content and text formats
Sorry, it did seem confusing when I read it back... So if I add some html to a text field, for example in a page or a block, whatever the text format of the field may be (raw html or basic...
Problems with HTML content and text formats
Could you please post clear, step by step instructions to reproduce the problem? Your first message refers to HTML content, and your second to text formats. Please add details to reproduce each...
Problems with HTML content and text formats