This isn’t too related to your main point, but every ordered field can be embedded into a field of Hahn series, which might be simpler to work with than surreals.
That page discusses the basics of Hahn series, but not the embedding theorem. (Ehrlich, 1995) treats things in detail, but is long and introduces a lot of definitions. The embedding theorem is stated on page 23 (24 in the pdf).
This isn’t too related to your main point, but every ordered field can be embedded into a field of Hahn series, which might be simpler to work with than surreals.
That page discusses the basics of Hahn series, but not the embedding theorem. (Ehrlich, 1995) treats things in detail, but is long and introduces a lot of definitions. The embedding theorem is stated on page 23 (24 in the pdf).