You're onto the right answer. Per Article II of the Constitution, the specific method for choosing Electors is up to each states' legislature, though Congress is empowered to set both the time of their selection and when they will vote for President.

Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors...

Up until 1820, direct appointment by the state legislatures was the most common method of choosing Electors, not through a popular election. Some states elected Electors via districts within the state. Some states used a combination of systems.

By the election of 1824, most states had adopted winner-take-all elections where one Presidential candidate receives all the state's Electors. This is not required by the Constitution; it was an political development to maximize the influence of a state in choosing the President.

1872 was the first year every state held a popular election to choose Electors. In 1876, Colorado was the last state to appoint Electors directly via the state Legislature.

There are two states (Nebraska, Maine) that do not apportion Electors on a winner-takes-all basis. In those states, one Elector is elected in each Congressional district, and two Electors are chosen based on the statewide vote total.

This feature of the Constitution - that conducting elections is a power delegated to the states - is how it was possible for enfranchisement to be extended to different groups at different times in different states. Each state decided for itself how elections occur and who gets to vote in them. For example, in Wyoming, women could vote in 1869 (then a U.S. territory, but enfranchisement continued when admitted as a state in 1890), 50 years before women's suffrage was extended to all states via the 19th amendment in 1919.