This Wikipedia article gives a pretty good writeup of the history.

The traditional method of legal education in the United States was an apprenticeship type model (and going back to England around the time when the current United States was English colonies), where aspiring lawyers would need to find apprenticeships/clerkships under the supervision and mentorship of existing lawyers.

Then during the 1800s, as you mention, schools like Harvard started offering legal education in the university environment, but university programs were competing with the rising professionalization and standardization of the professional schools where practitioners would formally take on many apprentices with a more formal, systematic curriculum of materials for the apprentices to study and do. So by 1900, the majority of lawyers still did not have law degrees from university-affiliated programs.

Around the same time, English universities were working through reforms of how legal education would work, and experimented with 6-year degrees that required no prior bachelor's degree, or 3-year programs that did require a separate bachelor's degree.

Throughout all of this, in England and in the United States, there was debate about whether it should be a postgraduate degree or a bachelor's degree, the extent to which the education should emphasize theory or practical skills, and how long these programs should be, and what the degree itself should be called.

Eventually, after fits and starts, all the different approaches started to merge into a new consensus in the United States by the 1960s, with the support of the American Bar Association: a 3-year Juris Doctor degree that requires a 4-year degree as a prerequisite. Apprenticeship has pretty much fallen by the wayside, with only very few states allowing that pathway as a method of becoming a lawyer, and some states (like California) requiring additional testing to prove competence (a first year exam known as the baby bar exam, then the bar exam), and almost all legal employers choosing not to hire any lawyers who don't have J.D.s, that anyone who actually wants to practice as an attorney in the United States will generally go to law school in the United States. Some states allow foreign lawyers to sit for the bar exam after only one additional year of education in an American law school (the international LLM), but that's a fairly niche path, mainly for people in New York who work in international things.