Learn English – the origin of the phrase “I’m game”

I don't know about the origins, but it is definitely not a recent expression. As the following examples found with Google Books indicate, the expression is used since the 19th century at least:

Pamphlets, Religious: Miscellaneous (1847)

It matters not if you call it a "fish pond" for fun and profit„or a five cent chance to get a quilt at a draw or a raffle given by a Ladies' Aid Society of Church, or the profits on editing a newspaper for one day, or running a railroad

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The American Angler (1897)

I am now fixing up some trout ponds, a pet scheme of mine, in which I shall propagate trout for fun and profit. The law here is, that no one can sell trout in market unless from private ponds, and I shall of course be in the game.

The phrase is a slight variant of "Is this the hill you want to die on?" which is often used in the military when discussing holding a position at all hazards. In this case, the answer is assumed to be "no".

When you decide to defend the spot to the limit, then "No better place to die" is often used. I have heard this used for many actions, back to the US Civil War and it probably was old during Ancient times.