On a more practical level, the schools that are good enough to land you a job as a doctor/lawyer are all so competitive that a hole in your CV like not having a bachelor's degree when everyone else has one is a real disadvantage.
Yes, I know that - but why have this requirement - as far as I know most other countries don't (certainly the UK doesn't).
As for why the bachelor's degree is required for medical schools, I don't know as much about that (I'm a lawyer and have studied the history of the US legal profession a bit). I suspect some of the reasons are the same (other than competition with the other profession, since I think the medical profession was the earlier mover here).
Edit: postwar -> post-WW2 as I realize most HN readers may not have grown up with that long-default US usage of "postwar".
Only a very small number of people do law as a second degree these days.
Amusingly, the process for training as an advocate is known as devilling - where you are apprenticed to a senior junior and during that time you are literally an advocate's devil :-)
Edit: Another thing we're starting to hear in conjunction with calls to reduce the 3-year program is to institutionalize something like the 'devilling' that you speak of, i.e. to replace the third academic year with an apprenticeship year.