To deal with superfluous carriage returns in TextEdit I want to let TextEdit never save &lt;CR&gt;, the carriage return. 

Different operating systems save end-of-line as different Unicode characters. 
For MS DOS (like Windows) it is &lt;CR&gt;&lt;LF&gt;, and for Unix and Mac &lt;LF&gt;, and for the very old Mac OS 9 just &lt;CR&gt;. 
Here &lt;CR&gt; is the carriage return, and &lt;LF&gt; the line feed. 

When sometimes I paste, into TextEdit, several paragraphs from a Windows-created file, for example pdf or doc/docx, &lt;CR&gt;'s are also copied. 
TextEdit does not fully conform with Unix standards, in that it does accept &lt;CR&gt;&lt;LF&gt;'s, by showing them indeed as line breaks. 
I cannot tell, then, whether a line break is Window-styled &lt;CR&gt;&lt;LF&gt; or Unix-styled &lt;LF&gt;. 

When I save my personal writings in pure text (txt) written in Markdown, I don't want these &lt;CR&gt;'s, but can neither see them, nor delete them in TextEdit. 
It is when I open the text file in question with Vim, did I spot those &lt;CR&gt;'s shown as ^M. 
Indeed, in Vim, it is easy to replace them with empty strings. 

However, I too edit in TextEdit often. 
Can I set TextEdit so that it never saves &lt;CR&gt;'s in a buffer of its? 
That is, when I paste a &lt;CR&gt; in a buffer, the &lt;CR&gt; is ignored and not written.
I don't believe that TextEdit has such an option.  It happily accepts all line ending types (even intermixed within the same file) and displays them properly.

You would probably need to use a text editor where you have direct control over the line endings and text encoding type when saving the file.  BBEdit is a great text editor that has such controls.

Another option is to convert the file with a utility like dos2unix.  The website doesn't offer a macOS binary download so you would need to either compile it yourself or install a package manager like Homebrew first then install the dos2unix package with brew install dos2unix. For an easy solution you can just run perl -pi -e 's/\r\n/\n/;' FILE.TXT as well.