the flesh of fish is often referred to as meat.
Yes and so is the flesh of a fruit. But that is in English. Is that true in Judaism as well? Also, according to your explanation the zemer “v’dagim” is superfluous.
Obviously someone who has health restrictions is unable to eat meat on Shabbos. But barring that, I don’t see any reason to not eat meat on Shabbos. Maybe if someone was disgusted by the taste. The Talmud says “There is no happiness except with meat and wine”.