I'd suggest along the lines of the next best option Joe proposes also would be the current JupyterLab and the document-centric Jupyter Notebook (versions 7 and up). Several of these you can use now without installing anything on your own system via MyBinder-served sessions, JupyterLite, py.cafe, or Anaconda Cloud. The first three listed don't necessarily involve you even signing up for anything or logging in. You can easily build in fancier UI's with ipywidgets, solarar, and anywidgets and use interactive dashboards/app providers already in the ecosystem, like Voila and Voici along with other. While I am mentioning interactive dashboards/ online apps, I'll add that py.cafe is targeted at this type of use and because Biopython is built in Pyodide, beyond use in JupyterLite, it should also be useful on py.cafe. Here is an example on Py.cafe where Biopython is used in support of a macromolecular visualization tool Molstar. (You can get to the editor to see the code by clicking the button in the upper right corner, or go directly to the page here.) That example isn't exactly what the original poster seeks because no UI elements are interacting with Biopython, but the expanded example here using two-way hover/highlight between ipymolstar and altair starts to give you an idea of what could be easily possible in a shareable form.
As for my first suggestion of Jupyter Notebook, I don't have an amazing example to point out off the top of my head but I do note that David Bikard's genome browser project. And I pointed out to the developer David Bikard that you could use it easily without installing anything of logging in because I forked the project and configured things so it launches via MyBinder-served sessions. You can see this in action by going here and clicking on the 'launch binder' badge you see here. If you are patient, you should get a session in your browser where 00_Examples.ipynb comes up and you can step through running the cells or just do Run > Run All Cells to really get a sense of what is possible.
(Oh yes, there was an effort to have a set of Jupyter .ipynb files for an introductory tutorial offered via MyBinder service here ; however, it hasn't been updated in five years. Sadly.)
Hopefully, adding some concrete-ish examples will provide a sense of the vast scope of options and possbilities already present and back up some of what Joe said in his first couple of paragraphs.
