I did a little looking around, and it seems that many Rishonim do hold it is a daily chiyuv min haTorah:
1) Rabbeinu Yonah in Iggeres Teshuva Ois 24 writes that one who doesn’t wear tefillin has close to 300 עונשים yearly.*
2) Smag (Asei #3) and Shibolei HaLeket Inyan Tefillin are clear that the Gemara (Menachos 44) which says that he who doesn’t fulfill the obligation of tefillin is oiver 8 miztvos refers to daily. (He proceeds to say some very interesting things about tefillin being beikar for reshaim, not tzaddikim… See also Maharik 174). — Courtesy of Google and קובץ בית אהרון וישראל, ר’ יעקב הכהן שבדרון*
3) Ritv”a Shabbos 49 – Chiyuv is Min HaTorah B’shas Krias Shema (Which is very interesting)
The Brisker Rav is brought by Reb Moshe Shternbuch תשובות והנהגות ח”א סי’ מ”ט, ח”ב סי’ ל, ח”ג סי” י”א.
* It is not muchach that these Rishonim contradict the Brisker Rav. It could be that although the chiyuv is not a new one every day, one who doesn’t wear them ever is punished for every day, like the Rambam writes by Milah.