A more common "entry level" position to gain a foothold in politics in urban areas is usually as a prosecutor. The career path is often to then run for DA ( whatever they call it ) of a largish city.
I'd trace this back to TR, the decline of machine politics and the rise of meritocratic city governance. Indeed, Barrack Obama offends many Conservatives because his background is as a community organizer, which harkens to their ear back to machine politics.
That's a peculiarity of the US system where the prosecution is heavily politicized and judges are elected. Other Western countries don't generally have that.
Politicians design policies. Engineers are good at working with complex systems. Seems like even better fit. We will just need some typewriters in suits to translate said based in science and reality policies into legalese.
I don't see why the west should be ruled by any particular type of profession, just engineers or just lawyers seem equally bad to me.
Honestly I think a lot of political disaster that is in the US could be traced back the the abundance of lawyers.
Seen from the outside, America to me is a country which has made every aspect of society into some form of legal struggle or court case. That just doesn't seem like a healthy way to organize society.
The polarized aspects of a court room, is reflected in American politics and public life itself. Politics is gridlocked because it is all about winning at any price. It is the logic of a lawyer.
I think this has become so ingrained in American phsycology that most Americans can't even see it anymore.
The flip side of the question is: Why do Western leaders generally lack STEM degrees?
Why should the US be ruled by lawyers?