Yes, and indeed that ability of surreal numbers to code arbitrary sets of ordinals plays a key role in my recent work on the first-order theory of Surreal Arithmetic, which is bi-interpretable with ZFC using this encoding.

Meanwhile, I don't really think of the epsilon numbers as a problem for the encoding of finite sets of ordinals by ordinals. After all, every ordinal is uniquely expressed as a sum of strictly descending powers of 2, and those exponents are the set that is coded. The issue with the epsilon numbers ε is that these ordinals are precisely the ordinals that encode their own singleton { ε }, precisely because 2^ε = ε.