1) To clarify - Rav Falk is dealing with laundry / bathing / haircuts. Not meat and wine. He is very explicit at the end of the teshuva
2) He only discusses non tzorchei Shabbos with regards to laundry. There's a strong sevara because of the takanos Ezra, that he brings in s"k #10 as well. I heard the same from other poskim as well.
3) Haircuts and rechitza lkovod shabbos he is matir as well
4) what he does not discuss is our question - where it's not lkovod shabbos. I'm going to shower tonight, but will shower again before shabbos. (same would apply to shaving).
5) There is no sevara in his teshuva that should be matir that.
As I noted, Rav falk's tshuva basically just addresses the question on when the heter of צורך שבת starts. Whereas the psak of the קדושת לוי was a blanket rule that no aveiles is noheg on Friday. However, Rav falk also agrees m'svora, in his words that I highlighted, that once a heter is given for צרכי שבת, that should apply for all of that sort even they're not for צרכי שבת, therefor he insists that all laundry should be permitted, and showering twice should be no different. But he's מסיים with a וצ׳׳ע since as he says he found no clear source for this. Therefore he's not giving a final heter on that part.
Now the קדושת לוי psak was actually on this exact question based on the same svura, with one clear distinction. The קדושת לוי posits that no aveiles were given on a Friday at all, Therefor even בשר ויין would me muter on Friday (but excluding Thursday night משום האבילות של יום), whereas rav falk who believes that aveiles was just removed for particular stuff that some of it's sort has a צורך שבת, then those stuff would have blanket heter even without being a צורך שבת, but in his view would בשר ויין be excluded.