The Or Zarua addresses this issue. I have the new MY edition on Hilchot Tisha b' Av. He goes into detail how "fowl" became prohibited DESPITE not triggering simchah as does meat from sacrificial type species.
He even says this Minhag was promoted to Issur.
V. 2 P. 574-575
«uldidan v'hoidna ha assur mishum bal "titosh torat imecha"»
Also See fn 14 citing SA O"Ch 552:1,2
Note: I believe someone says "Torat Imecha" is darshened as "Torat AMecha." Meaning it's not just Minhag Avot but rather the Minhag of the Q'hal at large, which would seem to include Geirim who join.
And this seems to make a genuine issur. I'm not sure if al Titosh creates a Hiyyuv to do something - EG Kapparot or Tashlich. But they can prohibit things BY Extension.