The wording I chose was “spirit of the Halacha” a poor translation of רוח ההלכה.
The reason is one should ask oneself – are we honoring this most lofty mitzvah of Tefillin in such a setting? Yes, there can be a problem with kevanah on the bracha and kevanah of the mitzvah. (See the Piskei siddur of the Alter Rebbe where there it says integrel kevanah one must have when donning. Tefillin. The Bach writes strong words in Orach Chayim siman 25 about one who does not have kevanah when donnig Tefillin.)
Another issue is can the Tefillin be properly donned kehalacha while someone is marching ? My assumption is this would present a great challenge. This matter is מעכב essential.
There can also be an issue with making a hefek (interruption) between the bracha and the act- potentially.

Certainly it would not be ideal to run into these challenges just because someone chooses to make their provocative statements to March with many vile individuals. At least to my opinion.
I will note that even if one chooses the approach of daring Mivtzaim activities, I don’t think they should all be posted in limelight of social media. Following the concept הלכה ואין מורין כן

You mention “the spirit of the halacha.” Do you have an actual source that there’s something disrespectful to the tefillin o put them on under such conditions, or is it simply your personal gut feeling?