countries most closely correlated with a single language family (roughly ranked)
Central African Republic, Ubangian (controversial classification as Niger-Congo)
Mongolian, Mongolic (point of diversity is in Mongolia, but most of the branches/subbranches are centered in Russia or China)
Australia, Pama-Nyungan (pre-contact; non-Pama-Nyungan was historically only spoken in a small part of the country)
Indonesia, Austronesian (while Taiwan is clearly the point of diversity for Austronesian, and there are several branches not spoken in Indonesia, i.e., Palauan, Chamorro, Polynesian, various Philippine branches... and there are Papuan languages spoken in Indonesia, Indonesia contains most Austronesian speakers and contains many Malayo-Polynesian branches)
India, Dravidian (~20% of the country speaks a Dravidian language, and the only language centered outside India is Brahui)
Thailand, Kra-Dai (~60% of speakers of languages in this family are Thai speakers, and 96% of Thailand speaks it as L1 or L2)
Sudan, Nilo-Saharan (This may be one of the most arbitrary. Assuming settlement of native ethnic groups was similar before Arab settlement, almost everyone in what is now Sudan spoke a language classified as Nilo-Saharan. Of course, Nilo-Saharan is a very controversial language family. Also, there were [controversial?] Niger-Congo speakers in the Kordofan/Nuba Mountains, and Beja on the Red Sea. Several few Nilo-Saharan branches aren't spoken in Sudan at all; Kunama, Nara, Surmic, Songhay and Kuliak. A few are barely spoken in the country, like Nilotic or Maban. There are so many holes to poke in this, but if you assumed the demographics of non-Arabs in the country would be directly extrapolated to 100% pre-contact, I think it would make the top 15 in the world in correlation between language family and political borders)
Korea, Koreanic (if it was a unified country)
Bougainville, Northern Bougainville & Southern Bougainville (It's hard to determine speaker counts for these languages; while the largest language in the hypothetical future country is Austronesian, these two Papuan [non-Austronesian] language families dominate the main island)
Guatemala, Mayan (Mamean, K'iche'an and Q'anjob'alan are centered in the country. Yucatecan, Huastecan and Ch'olan-Tzeltalan are not.)
Bolivia, Aymara (there are many language families with members in Bolivia, and isolates in Bolivia, but... about 80% of speakers are in Bolivia, and about 40% of indigenous language speakers in Bolivia speak Aymara)
Paraguay, Tupi-Guarani (While there are many minor Tupi-Guarani languages spoken outside of Paraguay, and several other language families and isolates spoken in Paraguay, the majority of people in Paraguay speak Guarani, there are still monolingual speakers, etc.)
Panama, Chibchan (pre-contact)
Uruguay, Charruan (pre-contact)
Namibia, Khoe-Kwadi (Kwadi was centered in Angola and Kalahari Khoe is centered in Botswana, but the majority of speakers of a Khoe language are Khoekhoe speakers, and 11% of people in Namibia speak Khoekhoe. Certainly not as close a correlation as in many of these countries)
East Timor, Timor-Alor-Pantar
In terms of US states, the following stick out:
Oklahoma, Caddoan (pre-contact; I know nomadic groups can be hard to pin down, apply that disclaimer to some of the items above, too)
New York, Iroquoian (there were also Algonquian languages spoken in New York, and Tuscarora, Nottoway and Cherokee were spoken further south, while Huron-Wyandot was spoken in Canada... please note that Lake Iroquoian was not the point of diversity for the family. This situation is a lot like Mongolia, with other branches being spoken outside of the state, and the sister branch, Huron-Wyandot, being spoken elsewhere, too)
Washington, Salishan (it's bizarre that anywhere on the west coast could be very closely correlated to a single language family, given the west coast is overall the most diverse area in North America, linguistically, by far. There are Chimakuan languages and a Wakashan language, Makah, spoken at the northern end of the Olympic peninsula. There are Chinookan and Sahaptian/Plateau Penutian languages spoken at the southern and eastern edges of the state. Kwalhoquia-Tlatskanai is a subbranch of Northern Athabaskan spoken in the state, too. And of course, Bella Coola and Tillamook are divergent branches of the family spoken outside of Washington, and there are Coast Salish languages in BC; the Interior Salish area also extends into BC, Idaho and Montana. However, probably at least 80% of land in Washington was settled by Salishan peoples at the time of contact)
A lot of this is really hard to quantify, but it's an interesting overlap of figures to consider.
