Altro can be either an adjective or a pronoun. It comes from Latin alter (also adjective and pronoun), meaning “the second of two”. In this case we're talking about the pronoun.
In Italian it has a more general usage, but always denotes some opposition between a single entity, which might be a group, and the rest: gli uni…, gli altri…, io…, gli altri. For instance
Metà di voi qua vadano,
e gli altri vadan là,
e pian pianin lo cerchino
(here Don Giovanni, disguised as Leporello, is dividing the group of peasants who are looking for him, so as to remain alone with Masetto and beat him).
In this usage the pronoun is almost always preceded by the article (exception: if a ne specifier precedes it).
When altro (masculine and singular) is not preceded by the article it is the same as the English else, so your conjecture that nient'altro can be used in place of non ad altro is correct. Here the opposition is between the game and everything else.
Note, however, that else needs some specifier, such as something, anything or nothing, whereas altro doesn't; here no specifier is needed, but it could be qualcosa in other contexts: ti serve altro? can also be ti serve qualcos'altro?. In this case, altro is “absolute” (corresponding to anything else), so no specifier.
The article in the Treccani dictionary is really clear about the matter.