"OK, the reason there's Democrats and Republicans Abroad is, they want Americans abroad to vote in the USA."
Not in the least.
As I knew the heads of both those Offices abroad, it's all about gaining influence and connections, not about getting Americans to vote.
Hell, I met both at the local Expat watering hole.... And hung out with them and their replacements for years...
In fact, they had no outreach whatsoever to Americans who lived or traveled there, none.
The objectives were always about making local connections, speaking at Universities, business conferences, local political parties, hosting parties and rubbing elbows with the local power brokers and influencers etc.