These type of seforim are extremely unreliable. I’ve seen in one of those type of Seforim that BR held there is no chiyuv to put on tefilin on a daily basis anymore.
This sefer is a collection of other printed seforim – obviously, some more reliable than others. You have to know the sefer he brings down and make sure it is reliable.
The example you mention here is also not as clear-cut as you’re making it out to be. What you [probably?] saw wasn’t a halachic psak that it is not necessary to wear tefillin. What it probably did say was something very different. Let me try guess.
There are seforim which discuss whether Tefillin is daily obligation, like Shema, or whether it is a chiyuv which is not necessarily timed daily. If someone missed a day of tefillin, was he mevatel an assei min haTorah? I have never seen the Brisker Rav’s position on the subject, but I can imagine him saying that Tefillin isn’t a daily chiyuv – which is, I think, the position of most Rishonim.
We do find that the ikker chiyuv of Tefillin is all day every day – not daily, but constant. Due to us not having a גוף נקי, we don’t wear it all the time. “The once a day” isn’t necessarily a daily chiyuv.
[I have never seen a source in Rishonim for a DAILY obligation to wear Tefillin. I’d be very grateful for someone to show me one.]
I can’t defend something I haven’t seen, and I certainly have found quotes from the Brisker Rav that are incorrect (On more than one occasion, I have found מפי שמועה the same diyuk in the same Rashi, with totally contradictory conclusions.)
Where is this quote from?