I can't comment on the popular discussion, but can offer a look at the Supreme Court decisions that eventually changed the situation.

"Eligible to serve" doesn't tell the entire story. New York and other states offered women the option not to serve as jurors, a privilege not offered to men. The Supreme Court decided in Fay v. New York (1947) that this practice violated neither the Due Process nor Equal Protection Clauses. Florida law required women who wished to be jurors to add their names to a special list; unsurprisingly, few did. In Hoyt v. Florida (1961), the Supreme Court let that stand, citing Fay. Justice Harlan wrote that, “[d]espite the enlightened emancipation of women from the restrictions and protections of bygone years, and their entry into many parts of community life formerly considered to be reserved for men, woman is still regarded as the center of home and family life.”

Finally, in Taylor v. Louisiana (1975), the Court ruled that “it is no longer tenable to hold that women as a class may be excluded or given automatic exemptions based solely on sex if the consequence is that criminal jury venires [panels] are almost totally male.”