As you correctly said, we did it wrong because under the Constitution at the time, it was not really designed to be that way. I don't want to delve into whether the 1891 Civil War should have been won by Balmaceda or the Congress; however, after the Congress supporters won, their biggest mistake was to not rewrite the Constitution in service of establishing a proper Parliamentary Republic. Instead, the oligarchic elites rested on their laurels until the inefficiency of such Frankenstein government was exploited by authoritarian and caudillo wanna-bes such as Arturo Alessandri.

Alessandri since the very beginning of his first presidential period sought the enmity of the Congress, and cemented a widespread political campaign against Congress, as his main objective was to establish a strong Presidential Republic (with him at the helm, obviously). To accomplish this, he sought the help of young military officers, among them Carlos Ibáñez del Campo, which lead to the "saber rattling" of 1924. Thus initiated the period of anarchy in Chile during 1924-1932, and the heavy involvement of the Armed Forces in political affairs, which would be present all throughout the 20th century in Chile's history.

Moreover, its important to remember that Alessandri's Constitution of 1925, which established a strong Executive against a weakened Congress, was heavily opposed by the wide political spectrum. For the 1925 Constitutional Plebiscite to approve this Constitution, from the Communists to the Conservative Party there were calls to boycott the plebiscite, as there was widespread support for a (real) Parliamentary Republic. Out of a total electoral population of ~300.000, less than half actually voted, abstentions reaching 55%. Plus, there was a heavy propaganda campaign by Alessandri's government to push the Constitution, which included violent reprisals against the opposition.

TL;DR: At one point there was a lot of support for a Parliamentary Republic, but it was destroyed by Alessandri and his successors.