Pakistan and India only (edit: ok, mainly) care about each other. The US and Britain are tightly aligned and they are separated from Eurasia by water, a non-trivial obstacle in its own right. The remaining Eurasian nuclear powers, Russia, China, and North Korea share borders. They are surrounded by essentially a skirt of nations. These nations are where proxy wars will occur. Various pretexts will develop from time to time.
The wars have not all been American. Russia has fought the Ukraine and the US continues to support Georgia (the country). China was heavily involved in Korea and Vietnam. The English Wikipedia is tracking 29 ongoing proxy wars, 21 of them are Eurasian (1).
Why are there so many proxy wars in Eurasia? First, ethnic violence. Racism is not uniquely American, and the slow migration of humanoids across Eurasia over history has left a lot of ethnic groups to quarrel with their neighbors. It is not uncommon that one of the major powers gets pulled in for some reason, and then other major powers see an advantage in supporting the opposing combatant.
Second, the major powers have always wanted buffer states like a chess player wants pawns. The US has Canada and Mexico, and a large number Central and South American states. China's empire has stretched from time to time as far as Okinawa and the Philippines to Mongolia, Korea, and brief intrusions into Japan proper. Russia has opposed the entry of the former Eastern Bloc states into NATO. Most recently, Russia has sold S-400 radars to Turkey, creating tension between NATO's newest and oldest members. Russia and China have both demonstrated they are willing to use armed force to extend their borders. Russia in Crimea, and China in the South China Sea (9-dash line). They have also demonstrated they will use economic means to advance their soft power (Russia creates dependencies through oil and gas trade, China floods various islands with money and cuts deals with the locals to "lease" their land (a trick they learned from the British, see Hong Kong).
The other battlefield is of course the Internet, ranging from intrusions and malicious behavior (e.g. stuxnet, N.K. vs Sony, Titan Rain) to open source information ops (Russian interference in the US 2016 election).
(1) https://en.wikipedia.org/wiki/List_of_proxy_wars#Ongoing_pro...